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Online math courses with Krista King.

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great at math.

I’ll help you make sense of math, one step at a time. Start wherever you are, from Pre-Algebra through Linear Algebra.

Every course for $39/month · 30-day money-back guarantee.

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20+ years: tutoring

“I have earned not one, but two As in the calculus courses I completed over the past year. I would have been happy with Bs, but you helped push me further and I feel like I really earned the As.”

Kevin H. · Back to calculus after 10 years

Find your course.

Start with the class you're taking now, or build the foundation that makes it easier.

See all 12 courses

Pre-Algebra

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Algebra 1

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Geometry

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Algebra 2

Try a lesson with me
Trigonometry

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Precalculus

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Probability & Statistics

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Calculus 1

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Calculus 2

Try a lesson with me
Calculus 3

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Differential Equations

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Linear Algebra

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Here’s how every lesson works.

Clear notes, a video of me explaining the concept and working an example, then practice questions to prove you’ve got it.

Preview your course

Start with the notes.

Every key idea and worked example you need, all in one place.

RoundingPre-Algebra

Open the full notesShow only page one

Dividing polynomialsAlgebra 1

Open the full notesShow only page one

Scalene, isosceles, and equilateral trianglesGeometry

Scalene, isosceles, and equilateral triangles, page 1 of 5

Open the full notesShow only page one

Scalene, isosceles, and equilateral triangles, page 2 of 5

Scalene, isosceles, and equilateral triangles, page 3 of 5

2. If exactly two sides are equal, classify the triangle as isosceles.
If all three sides are equal, go to Step 3.

Scalene, isosceles, and equilateral triangles, page 4 of 5

Scalene, isosceles, and equilateral triangles, page 5 of 5

Graphing circlesAlgebra 2

Open the full notesShow only page one

Sine, cosine, and tangentTrigonometry

Open the full notesShow only page one

Polar coordinatesPrecalculus

Open the full notesShow only page one

For r > 0, use the signs of x and y to find the angle θ.
At the origin, r = 0 and any angle works.

Box-and-whisker plotsProbability & Statistics

Open the full notesShow only page one

Idea of the limitCalculus 1

Open the full notesShow only page one

The limit is 6 because f(x) = x + 1 is continuous at x = 5. Its values approach 6 from both sides, and f(5) = 6. In limit notation, here’s how that looks:

Since √x + 2 is continuous at 16, substitute x = 16.

Over and underestimationCalculus 2

Open the full notesShow only page one

Estimate the area under f(x) = √(x + 5) on [−5, 20]. Use five unequal subintervals with endpoints where x + 5 is a perfect square. Choose sample points that give the largest estimate for this partition.

The function increases on [−5, 20], so left endpoints underestimate the area and right endpoints overestimate it. Right endpoints give the largest sum for these subintervals.

Choose endpoints where x + 5 equals 0, 1, 4, 9, 16, or 25. Then the heights √(x + 5) are 0, 1, 2, 3, 4, and 5.

Direction cosines and direction anglesCalculus 3

Direction cosines and direction angles, page 1 of 4

Open the full notesShow only page one

Direction cosines and direction angles, page 2 of 4

Direction cosines and direction angles, page 3 of 4

Direction cosines and direction angles, page 4 of 4

Autonomous equations and equilibrium solutionsDifferential Equations

Autonomous equations and equilibrium solutions, page 1 of 12

So far we’ve looked at linear and nonlinear first-order ordinary differential equations. Now we’ll focus on autonomous equations. They can be linear or nonlinear.

Where f(y) ≠ 0, we can separate variables. Check any zeros of f(y) separately for equilibrium solutions.

Open the full notesShow only page one

Autonomous equations and equilibrium solutions, page 2 of 12

Autonomous equations and equilibrium solutions, page 3 of 12

Autonomous equations and equilibrium solutions, page 4 of 12

Autonomous equations and equilibrium solutions, page 5 of 12

Autonomous equations and equilibrium solutions, page 6 of 12

Autonomous equations and equilibrium solutions, page 7 of 12

Autonomous equations and equilibrium solutions, page 8 of 12

Autonomous equations and equilibrium solutions, page 9 of 12

Autonomous equations and equilibrium solutions, page 10 of 12

Autonomous equations and equilibrium solutions, page 11 of 12

Autonomous equations and equilibrium solutions, page 12 of 12

Equation of a plane, and normal vectorsLinear Algebra

Equation of a plane, and normal vectors, page 1 of 7

A plane is determined by a point on it and a nonzero normal vector. For any two points in the plane, the vector connecting them is orthogonal to the normal. A plane can pass through the origin or be shifted away from it.

Open the full notesShow only page one

Equation of a plane, and normal vectors, page 2 of 7

Equation of a plane, and normal vectors, page 3 of 7

Equation of a plane, and normal vectors, page 4 of 7

Equation of a plane, and normal vectors, page 5 of 7

Equation of a plane, and normal vectors, page 6 of 7

Equation of a plane, and normal vectors, page 7 of 7

Multiplying the equation by any nonzero scalar gives another equation for the same plane. Its coefficients give another normal vector. Positive scalars keep the same direction, and negative scalars reverse it.

Watch me explain the topic.

We’ll start with the big idea and work through an example together.

RoundingPre-Algebra

Watch the lesson in the course

Dividing polynomialsAlgebra 1

Watch the lesson in the course

Scalene, isosceles, and equilateral trianglesGeometry

Watch the lesson in the course

Graphing circlesAlgebra 2

Watch the lesson in the course

Sine, cosine, and tangentTrigonometry

Watch the lesson in the course

Polar coordinatesPrecalculus

Watch the lesson in the course

Box-and-whisker plotsProbability & Statistics

Watch the lesson in the course

Idea of the limitCalculus 1

Watch the lesson in the course

Over and underestimationCalculus 2

Watch the lesson in the course

Direction cosines and direction anglesCalculus 3

Watch the lesson in the course

Autonomous equations and equilibrium solutionsDifferential Equations

Watch the lesson in the course

Equation of a plane, and normal vectorsLinear Algebra

Watch the lesson in the course

Try a question, then see the full solution.

Find out what you know and what to practice next.

RoundingPre-Algebra

Round to the nearest hundredth.

0.73862

0.75

0.72

0.74

0.73

You got it!

Try again.

See the worked solution

Dividing polynomialsAlgebra 1

Simplify the expression.

(x^2+x+8)\div\left(x-1\right)

x+\frac{8}{x-1}

x^2+x+4

x^2

x+2+\frac{10}{x-1}

You got it!

Try again.

See the worked solution

Dividing polynomials worked solution, page 1 of 1

Scalene, isosceles, and equilateral trianglesGeometry

Choose the value that can be the measure of a base angle of an isosceles triangle.

1^\circ

90^\circ

98^\circ

100^\circ

You got it!

Try again.

See the worked solution

Scalene, isosceles, and equilateral triangles worked solution, page 1 of 1

Graphing circlesAlgebra 2

Which circle is the graph of

(x-2)^2+(y+3)^2=4

ABCD

You got it!

Try again.

See the worked solution

Graphing circles worked solution, page 1 of 1

Sine, cosine, and tangentTrigonometry

In the right triangle

ABC

\cos \angle{C}=3/5

. If the adjacent side has length

30

, what is the length of the hypotenuse?

30

50

\frac{1}{50}

40

You got it!

Try again.

See the worked solution

Sine, cosine, and tangent worked solution, page 1 of 1

Polar coordinatesPrecalculus

What are the rectangular coordinates

(x,y)

of the point that has polar coordinates

(r,\theta)=(4,\pi)

(x,y)=(4,0)

(x,y)=(0,-4)

(x,y)=(0,4)

(x,y)=(-4,0)

You got it!

Try again.

See the worked solution

Polar coordinates worked solution, page 1 of 1

Box-and-whisker plotsProbability & Statistics

The box plot shows the number of hours slept on vacation. What is the median of the data?
Box plot of hours slept on vacation.

2

6

7.5

8.5

You got it!

Try again.

See the worked solution

Box-and-whisker plots worked solution, page 1 of 1

Idea of the limitCalculus 1

Evaluate the limit.

\lim_{x\to3}\frac{x-6}{x}

-3

3

-1

1

You got it!

Try again.

See the worked solution

Idea of the limit worked solution, page 1 of 1

Over and underestimationCalculus 2

Use Riemann sums to find the smallest and largest estimations of the area under

f(x)

[1,6]

with

n=5

subintervals.

f(x) = \frac{1}{8}{x^2} - \frac{1}{4}x + \frac{1}{2}

Smallest area estimation of

48/8

and largest of

73/8

Smallest area estimation of

23/4

and largest of

37/4

Smallest area estimation of

45/8

and largest of

35/4

Smallest area estimation of

35/4

and largest of

45/8

You got it!

Try again.

See the worked solution

Over and underestimation worked solution, page 1 of 3

Over and underestimation worked solution, page 2 of 3

Over and underestimation worked solution, page 3 of 3

Direction cosines and direction anglesCalculus 3

Find the direction cosines of

a=\langle5,6\rangle

\cos{\alpha}=\frac{5}{\sqrt{61}}

and

\cos{\beta}=\frac{6}{\sqrt{61}}

\cos{\alpha}=\frac56

and

\cos{\beta}=\frac65

\cos{\alpha}=\frac{5}{\sqrt{11}}

and

\cos{\beta}=\frac{6}{\sqrt{11}}

\cos{\alpha}=\frac{5}{\sqrt{121}}

and

\cos{\beta}=\frac{6}{\sqrt{121}}

You got it!

Try again.

See the worked solution

Direction cosines and direction angles worked solution, page 1 of 1

Autonomous equations and equilibrium solutionsDifferential Equations

Find any equilibrium solutions of the autonomous differential equation, then determine whether each solution is stable, unstable, or semi-stable.

\frac{dy}{dt} =4y-4y^{2}

y=0

is a stable solution and

y=1

is an unstable solution

y=0

is an unstable solution and

y=1

is a stable solution

y=0

and

y=1

are unstable solutions

y=0

and

y=1

are stable solutions

You got it!

Try again.

See the worked solution

Autonomous equations and equilibrium solutions worked solution, page 1 of 2

Autonomous equations and equilibrium solutions worked solution, page 2 of 2

Equation of a plane, and normal vectorsLinear Algebra

What is the normal vector to the plane?

3x+5y+9z=-26

\vec{n}=(3,5,9)

\vec{n}=(-3,-5,9)

\vec{n}=(3,-5,-9)

\vec{n}=(-3,5,-9)

You got it!

Try again.

See the worked solution

Equation of a plane, and normal vectors worked solution, page 1 of 1

Extra practicePre-AlgebraAlgebra 1GeometryAlgebra 2TrigonometryPrecalculusProbability & StatisticsCalculus 1Calculus 2Calculus 3Differential EquationsLinear Algebra

Walk into test day ready.

Extra practice for every topic, plus a final exam to show you’ve got it.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Worksheets
Review the key ideas and try a few problems.
Practice workbooks
Extra problems, with solutions to check your work.
Final exams
A practice run before test day.
Formula sheets
Every key formula, in one quick reference.

Personal supportPre-AlgebraAlgebra 1GeometryAlgebra 2TrigonometryPrecalculusProbability & StatisticsCalculus 1Calculus 2Calculus 3Differential EquationsLinear Algebra

You’re never on your own.

Real help, right when you need it.

KristaYour teacher

Krista²AI math tutor

I teach every lesson. If you get stuck, Krista², my AI tutor, is there day or night. She’ll guide you through the thinking step by step. And, of course, you can always reach out to me directly. 🤓

Krista²

For −1 + 8, I got −9 because I added 1 and 8 and kept the minus sign. What am I doing wrong?

The minus sign belongs to the $1$ , not to the whole sum. Because $-1$ and $8$ have opposite signs, they partly cancel rather than combining into a larger negative number.

Read the full replyShow less

Think of a number line: start at $-1$ . Adding positive $8$ means moving 8 units to the right. Where do you land?

Example conversation · Pre-Algebra

Krista²

I’m solving 3(x − 4) = 2x + 5. I expanded it to 3x − 4 = 2x + 5, then got x = 9. It fails when I check it. Where did I mess up?

The first error is in the expansion. The $3$ multiplies both terms inside the parentheses, including $-4$ :

3(x-4)=3x-12.

Read the full replyShow less

So your equation becomes

3x-12=2x+5.

Your later algebra was consistent with the expansion you wrote; it’s that expansion that changed the equation. What do you get if you subtract $2x$ from both sides of the corrected equation?

Example conversation · Algebra 1

Krista²

A triangle has an exterior angle of 125° and one of the other interior angles is 18°. I did 180 − 125 − 18 = 37° for the missing angle. Is my setup right?

Not quite: your setup uses $125^\circ$ as though it were an interior angle, but it’s an exterior angle.

Read the full replyShow less

An exterior angle equals the sum of the two interior angles that aren’t next to it. So if $x$ is the missing angle, we have

x+18^\circ=125^\circ.

What operation would you use to isolate $x$ ?

Example conversation · Geometry

Krista²

I rewrote log₈(x) = 4 as x = 8 × 4 = 32. My notes say that’s wrong. What did I mix up?

You mixed up an exponent with multiplication. A logarithm tells us what exponent to put on the base.

\log_8(x)=4

Read the full replyShow less

means “8 raised to what power equals $x$ ?” The logarithm says that power is 4, so the correct rewrite is

x=8^4.

That means multiplying four factors of 8, rather than multiplying 8 by 4. Can you evaluate $8^4$ ?

Example conversation · Algebra 2

Krista²

I know cos C = 3/5 and the adjacent side is 30. I multiplied 30 by 3/5 and got 18 for the hypotenuse. Why is that wrong?

Your multiplication is correct, but the ratio is being used in the wrong direction. Cosine means

\cos C=\dfrac{\text{adjacent}}{\text{hypotenuse}}.

Read the full replyShow less

So $\cos C=3/5$ tells us the adjacent side is $3/5$ of the hypotenuse, not the other way around. With adjacent side $30$ , we have

\dfrac{3}{5}=\dfrac{30}{h},

or equivalently,

30=\dfrac{3}{5}h.

Also, $18$ can’t be the hypotenuse because the hypotenuse must be longer than the adjacent side, $30$ .

What operation would undo multiplying $h$ by $3/5$ ?

Example conversation · Trigonometry

Krista²

For the polar point (4, π), I used x = 4 because r is 4, and y = 0 because π is on the x-axis. Why isn’t (4, 0) correct?

You’re right that $y=0$ , because the angle $\pi$ points along the x-axis. The key is which direction along that axis.

Read the full replyShow less

In polar coordinates, $r=4$ means move 4 units in the direction given by the angle. Since $\pi$ radians is $180^\circ$ , that direction is left, along the negative x-axis. So $r=4$ doesn’t mean $x=4$ .

Using the conversion formulas:

x=r\cos\theta=4\cos\pi=-4

y=r\sin\theta=4\sin\pi=0.

The rectangular point is therefore $\boxed{(-4,0)}$ : 4 units left of the origin.

Example conversation · Precalculus

Krista²

A normal distribution has mean 150 and standard deviation 28. For 94 to 206 I used 68% because it’s one interval. What am I missing?

The key is how many standard deviations the endpoints are from the mean, not how many intervals are written.

Read the full replyShow less

Both endpoints are $56$ units from $150$ . Since the standard deviation is $28$ ,

\dfrac{56}{28}=2.

So $94$ to $206$ covers two standard deviations below and above the mean. The $68\%$ rule applies within one standard deviation of the mean.

According to the empirical rule, what percentage falls within two standard deviations?

Example conversation · Probability & Statistics

Krista²

I canceled the x’s in (x − 6)/x and got 1 − 6 = −5. Is that how I find the limit as x approaches 3?

Not quite. Canceling works for factors multiplying the whole numerator and denominator. In $x-6$ , the $x$ is only one term, so we can’t cancel it out of the whole numerator.

Read the full replyShow less

Instead, divide each numerator term by $x$ :

\dfrac{x-6}{x} =\dfrac{x}{x}-\dfrac{6}{x} =1-\dfrac{6}{x} \qquad (x\ne 0).

Since the denominator isn’t zero at $x=3$ , we can find the limit by substitution. What do you get when you substitute $3$ for $x$ in this corrected expression?

Example conversation · Calculus 1

Krista²

For ∫ x cos x dx, I chose u = cos x and dv = x dx. Now the integral is getting harder. How do I choose better?

Your choice is valid, but differentiating $\cos x$ keeps producing trig functions, while integrating $x$ raises its degree. That makes the remaining integral harder.

Read the full replyShow less

For integration by parts, aim to choose $u$ so it gets simpler when differentiated, and $dv$ so it’s easy to integrate. Here, a better choice is

u=x,\qquad dv=\cos x\,dx.

Then

du=dx,\qquad v=\sin x.

Using the integration-by-parts formula,

\int u\,dv=uv-\int v\,du,

we get

\int x\cos x\,dx=x\sin x-\int\sin x\,dx.

Now the remaining integral is simpler because the factor $x$ is gone. What antiderivative would you use for $\sin x$ ?

Example conversation · Calculus 2

Krista²

For f(x, y) = 4x³y², I found ∂f/∂y = 12x² · 2y. Why is that wrong?

When taking a partial derivative with respect to $y$ , we treat $x$ as a constant. So $4x^3$ stays unchanged, and only $y^2$ is differentiated:

\dfrac{\partial f}{\partial y} =4x^3\cdot 2y =8x^3y.

Read the full replyShow less

Your derivative of $y^2$ , which is $2y$ , is correct. The error is changing $4x^3$ into $12x^2$ : that would be differentiating it with respect to $x$ , but here we’re differentiating with respect to $y$ .

Example conversation · Calculus 3

Krista²

For dy/dx = sec²x/tan²y, I divided by tan²y and got dy/tan²y = sec²x dx. Am I separating this correctly?

Not quite. Since $\tan^2 y$ is in the denominator on the right, we multiply both sides by $\tan^2 y$ to cancel it, then multiply by $dx$ :

\tan^2 y\,dy=\sec^2 x\,dx.

Read the full replyShow less

Now all the $y$ -terms are on the left and all the $x$ -terms are on the right, so the variables are separated.

Can you rewrite $\tan^2 y$ using the identity $1+\tan^2 y=\sec^2 y$ to prepare the left side for integration?

Example conversation · Differential Equations

Krista²

For R₁ ↔ R₂, I swapped just the first numbers in the two rows. My answer is wrong. What does that symbol tell me to do?

The symbol $R_1 \leftrightarrow R_2$ means swap the entire first and second rows, not just their first entries. Every entry moves with its row, including the number after the divider in an augmented matrix.

Read the full replyShow less

For example,

\begin{bmatrix} 1 & 2 & \mid & 7\\ 3 & 4 & \mid & 8 \end{bmatrix} \quad\longrightarrow\quad \begin{bmatrix} 3 & 4 & \mid & 8\\ 1 & 2 & \mid & 7 \end{bmatrix}.

Try your swap again, moving each whole row into the other row’s position.

Example conversation · Linear Algebra

What students have gone on to do.

Some were in high school. Some came back to math decades later. Here’s where they ended up.

“Watching her videos did much more than teach me how to work problems. They made the proofs my professors did in class make sense. My grades went from C in Calc 1, to B+ in Calc 2, to A's in Calc 3 and Linear Algebra.”

Holly M.Calculus 1 through Linear Algebra

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Lawrence D.Returned to math at 34

“I took calculus for the first time at the age of 39. I passed Calc 1 with a 92, and I passed Calc 2 with a 95. Now I am chugging through Calc 3 and physics, which I have changed my major to. I could not have done this without your courses.”

Jennifer L.Started calculus at 39

“I'm now enrolled in Calc 3. My professor is astounded as to how well I'm doing. At one point during Calc 2 I thought about dropping math entirely, but since joining I've added a math minor. You've given me so much confidence in my mathematics ability.”

TreyCalculus 2 and 3

“I chose Ms. King's Algebra course as a review to move past my previous level and must say that I not only enjoy the course she offers but am getting more from it learning wise than I could have imagined. I wish to complete all of her classes and am extremely confident that her teaching style can get me there.”

Dean F.Algebra 1

Hi, I’m Krista.

I spent a lot of time tutoring math when I was a student at the University of Notre Dame (Go Irish! 🍀), and I loved helping people figure out calculus. I’ve been tutoring for over 20 years now, and I teach these courses the same way I would if we were sitting down together.

Whether you’re aiming for an A in your class or coming back to math after years away, I’d love to help you get there.

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Message Krista with your questions

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Message Krista with your questions

Start yearly

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A few things you
might be wondering…

Does one membership include every course?

Yes. Both plans include all 12 courses, from Pre-Algebra through Linear Algebra, plus lessons, study resources, and support. You can move between courses whenever you need to.

Who are the courses for?

Students taking a math class in high school or college, adults coming back to math after time away, and parents looking for help for their student. The courses run from Pre-Algebra through Linear Algebra, so you can start wherever you are.

I'm a parent. How does this work for my child?

Once you choose a membership, help your child set up their course account and pick the course that matches what they’re learning. If you’re unsure where to start, try the placement quiz together. Your child can work through the lessons at their own pace. If they get stuck, Krista² asks questions and gives hints to help them think through the problem. If you have questions about setup, course choice, or your child’s progress, send me a message.

Can I use this alongside my current class?

Yes. Follow a course from the start, or jump to the topics you're covering in class. You can also send me your syllabus or assignments and I'll build a custom learning path that follows your class.

What if I don't know where to start?

Take the Find your starting point quiz. It checks what you already know and shows you the exact lesson to start with. You can also browse the course outlines to find the topics you need.

What if I need help with a lesson?

Krista² is available inside your lessons 24/7. It’s an AI tutor trained in my teaching style. You can also message me for personal support.

How do cancellation and the guarantee work?

You can cancel from your account settings and keep access through the end of your billing period. If the courses aren’t the right fit, contact me within 30 days for a refund under the money-back guarantee.

Can I try your teaching before I join?

Of course. Open a free course preview, or spend some time with my blog and YouTube lessons. You’re welcome to keep learning while you decide.

You can begreat at math.

Find your course.

Pre-Algebra

Algebra 1

Geometry

Algebra 2

Trigonometry

Precalculus

Probability & Statistics

Calculus 1

Calculus 2

Calculus 3

Differential Equations

Linear Algebra

Not sure where to start?

Here’s how every lesson works.

Start with the notes.

Watch me explain the topic.

Try a question, then see the full solution.

Walk into test day ready.

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Worksheets

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Final exams

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Final exams

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Practice workbooks

Final exams

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Final exams

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Final exams

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You’re never on your own.

What students have gone on to do.

Hi, I’m Krista.

One membership, every course.

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A few things youmight be wondering…

Courses

You can be
great at math.

A few things you
might be wondering…