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Digital SAT · Solve it in Desmos

How to Solve Quadratic Equations on Desmos (Digital SAT)

For nearly every numeric quadratic, graphing beats the algebra and skips sign and radical errors. Save the quadratic formula for unknown coefficients, or use the discriminant when the question is about how many solutions there are.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for quadratic equations

The Desmos steps

  1. 1

    Type it as a function

    Enter y = x^2 - 6x, say, without factoring or rearranging first.

  2. 2

    Click the zeros

    The x-axis crossings are the solutions, and Desmos labels their coordinates.

  3. 3

    Click the vertex for a max or min

    For an extreme value or a count, read the vertex or how often the curve meets the axis, then answer exactly.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Easy example
What is the positive solution to the equation x2−6x=0x^{2} - 6x = 0?
A
−6-6
B
00
C
33
66

A: Incorrect. This flips the sign; factoring gives x(x−6)=0x(x-6)=0, so the positive root is 6.

B: Incorrect. This is the other solution, but 0 is not positive; the positive solution is 6.

C: Incorrect. This divides 6 by 2 instead of reading the root from x−6=0x - 6 = 0.

D: Correct. Factoring gives x(x−6)=0x(x-6)=0, so the solutions are 0 and 6, and the positive one is 6.

Explanation

Factor x2−6x=x(x−6)x^{2} - 6x = x(x-6); the roots are 0 and 6, so the positive solution is 6.

Medium example
What is the positive solution to the equation x2−2x−35=0x^{2} - 2x - 35 = 0?
A
−7-7
B
−5-5
C
55
77

A: Incorrect. This flips the sign of the positive root; factoring gives (x−7)(x+5)=0(x-7)(x+5)=0, so the positive root is 7.

B: Incorrect. This is the negative root; the positive solution is 7, not −5-5.

C: Incorrect. This flips the sign of the negative root −5-5 instead of taking the positive root 7.

D: Correct. Factoring gives (x−7)(x+5)=0(x-7)(x+5)=0, so the solutions are 7 and −5-5, and the positive one is 7.

Explanation

Factor x2−2x−35=(x−7)(x+5)x^{2} - 2x - 35 = (x-7)(x+5); the roots are 7 and −5-5, so the positive solution is 7.

Hard example
The equation 3x2−12x=03x^2-12x=0 has two solutions. What is the nonzero solution?
A
−4-4
B
00
C
33
44

A: Incorrect. (x−4)=0(x-4)=0 gives x=4x=4, not −4-4.

B: Incorrect. 0 is the other (zero) solution.

C: Incorrect. 3 is the leading coefficient, not a solution.

D: Correct. 3x2−12x=3x(x−4)3x^2-12x=3x(x-4), so the solutions are 0 and 4; the nonzero one is 4.

Explanation

3x(x-4)=0 gives x = 0, 4; nonzero is 4.

Try it with the calculator

Question 1easy

What is the greater of the two solutions to the equation x2−7x+12=0x^{2} - 7x + 12 = 0?

Question 2easy

What is the positive solution to the equation x2+2x−15=0x^{2} + 2x - 15 = 0?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

Start the free diagnostic

Desmos questions

Can Desmos solve quadratic equations questions on the SAT?

For nearly every numeric quadratic, graphing beats the algebra and skips sign and radical errors. Save the quadratic formula for unknown coefficients, or use the discriminant when the question is about how many solutions there are.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master quadratic equations, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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