Perfect1600

Digital SAT · Quadratic & Exponential Functions

How to Solve Quadratic Equations on the Digital SAT

A quadratic carries an x-squared term and usually has two solutions, the x-values that make it zero. Three roads lead there. Factor when the numbers are friendly, reach for the quadratic formula when they are not, or graph the equation in Desmos and click where it crosses the x-axis. Then read the last line closely, because it may want only the positive solution, the sum of the roots, or how many solutions there are.

Written from Perfect1600’s analysis of every quadratic equations question in our bank·Method checked against the current Bluebook test
per test
3 per test
per test
typical difficulty
Medium to hard
typical difficulty
practice questions
111
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

91%
Graph and read the zeros

Nine in ten are solved by graphing the function and clicking the x-axis crossings.

47%
Grid-ins

Almost half are typed answers, so a dropped root or sign has nothing to catch it.

watch
Answer the exact root

Many want only the positive solution or the sum, and the other root is offered as bait.

How to recognize quadratic equations questions

  • The equation has an x-squared term, or the question names a parabola, and asks for a solution, a zero, or a count.
  • It may ask for the positive solution, the sum of the roots, or a feature like the vertex.
  • Choices are single numbers, or the phrases no solution, exactly one, and infinitely many.
  • Almost half are grid-ins, so you produce the value yourself.

Why students miss these

The algebra is only half the story; the points slip away at the edges. A sign goes wrong under the radical, the plus-or-minus gets dropped, or a student answers with every root when only the positive one was wanted. With nearly half of these as grid-ins, there is no choice to catch the slip. Graphing removes the arithmetic, but you still have to click the right feature, a zero for a solution, the vertex for a max or min.

The step-by-step method

  1. 1

    Set it equal to zero

    Move every term to one side so the equation reads (quadratic) = 0; the solutions are the x-values that make it true.

  2. 2

    Factor first if you can

    If it factors cleanly, set each factor to zero. Each gives a solution, faster than the formula for small numbers.

  3. 3

    Otherwise use the formula

    Substitute a, b, and c carefully and keep the plus-or-minus, watching the sign of b and the term under the radical.

  4. 4

    Answer exactly what is asked

    Reread for the positive solution, the sum, the count, or a feature like the vertex, not the full list of roots.

Solving it on Desmos

  1. Type it as a function. Enter y = x^2 - 6x, say, without factoring or rearranging first.
  2. Click the zeros. The x-axis crossings are the solutions, and Desmos labels their coordinates.
  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.

Full Desmos walkthrough for quadratic equations→

When to use it: 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.
See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

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.

The common traps

PatternWhat it doesThe tell
Answered every rootGave both solutions, or the wrong one, when only the positive root or the sum was wanted.Your value is a real solution, but not the specific one the last line asked for.
Sign slip in the formulaA wrong sign on minus-b or under the radical flips the result.The answer is close but off by a sign, and a grid-in will not warn you.
Dropped the plus-or-minusReported one root and missed the second.A both-solutions or count prompt exposes the missing root.
Wrong feature on the graphRead the vertex when a zero was asked, or the reverse.An x-intercept is a solution; the vertex is a max or min.

Try it: two real questions

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.

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Often confused with

Linear Equations & Inequalities
Linear Equations
Quadratic & Exponential Functions
Quadratic-Linear Systems
Equivalent Expressions
Equivalent Expressions

Related reading

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Common questions

How do you solve a quadratic equation on the SAT?

Set it to zero, then factor if the numbers are clean or use the quadratic formula if not. On the digital test, graphing the function in Desmos and clicking the x-axis crossings is usually the fastest route.

Can I use Desmos to solve quadratics on the Digital SAT?

Yes, and it is usually quickest. Type the equation as y = ..., then click the x-intercepts for the solutions or the vertex for a max or min. It skips the sign and radical errors that cost grid-in points.

Do I still need the quadratic formula on the Digital SAT?

Now and then, for equations with unknown coefficients and for number-of-solutions questions where the discriminant decides. For ordinary numeric quadratics, graphing is faster and safer.

How do I find the number of solutions of a quadratic?

Look at the discriminant, the value under the radical. Positive means two real solutions, zero means one, negative means none. On a graph, it is how often the parabola meets the x-axis.

What is the fastest way to solve quadratics on the SAT?

Graph the function in Desmos and click the feature you need. Factoring is fast for clean numbers, but graphing works on nearly every quadratic and avoids mistakes, which matters since almost half are grid-ins.

Practice quadratic equations the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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