Quadratics on the Digital SAT: Factoring, Formula, Vertex & Desmos

Master Digital SAT quadratics: factoring, roots, quadratic formula, discriminant, vertex form, graph interpretation, Desmos strategy, and original practice.

Published 15 September 2026

Quadratics belong to Advanced Math, one of the two largest SAT Math domains.

College Board describes Advanced Math as including quadratic, exponential, polynomial, rational, radical, absolute-value, and other nonlinear relationships. So you should expect nonlinear algebra overall—but no official source guarantees a fixed number of quadratic questions.

Recognize the three useful forms

Standard form

[ y=ax^2+bx+c ]

Useful for:

quadratic formula;
discriminant;
y-intercept (c).

Factored form

[ y=a(x-r_1)(x-r_2) ]

Useful for:

roots / x-intercepts (r_1,r_2).

Vertex form

[ y=a(x-h)^2+k ]

Useful for:

vertex ((h,k));
minimum/maximum;
graph transformations.

A major SAT skill is choosing the form that exposes the information you need.

Factoring

Example 1

Solve:

[ x^2-7x+12=0 ]

Find two numbers that multiply to 12 and add to -7:

-3 and -4.

[ (x-3)(x-4)=0 ]

So:

[ x=3,\ 4 ]

Difference of squares

[ x^2-25=0 ]

[ (x-5)(x+5)=0 ]

[ x=\pm5 ]

Recognizing structure is faster than using the quadratic formula.

Quadratic formula

For:

[ ax^2+bx+c=0 ]

[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} ]

Example 2

[ 2x^2+x-6=0 ]

(a=2,\ b=1,\ c=-6).

[ x=\frac{-1\pm\sqrt{1-4(2)(-6)}}{4} ]

[ x=\frac{-1\pm\sqrt{49}}{4} ]

[ x=\frac{-1\pm7}{4} ]

Solutions:

[ x=\frac32,\ -2 ]

This one also factors, but the formula always gives a systematic route.

The discriminant

The expression:

[ b^2-4ac ]

tells you how many real solutions a quadratic has.

(>0): two distinct real solutions
(=0): one repeated real solution
(<0): no real solutions

Example 3

For what value of (k) does

[ x^2-6x+k=0 ]

have exactly one real solution?

Set discriminant to zero:

[ (-6)^2-4(1)(k)=0 ]

[ 36-4k=0 ]

[ k=9 ]

Vertex form

Example 4

[ f(x)=2(x-3)^2-5 ]

Vertex:

[ (3,-5) ]

Because (a=2>0), the parabola opens upward.

Minimum value:

[ -5 ]

The SAT may ask for:

minimum/maximum;
x-value at the vertex;
transformation;
a parameter.

Completing the square

Suppose:

[ x^2-8x+11 ]

Complete the square:

[ x^2-8x+16-5 ]

[ (x-4)^2-5 ]

Now the vertex is visible immediately:

[ (4,-5) ]

Intercepts

y-intercept

Set (x=0).

For:

[ y=3x^2-4x+7 ]

y-intercept = 7.

x-intercepts

Set (y=0) and solve.

These are the roots.

Desmos strategy

Desmos can make quadratics fast.

Find roots

Graph:

[ y=x^2-7x+12 ]

Click the x-intercepts.

Find vertex

Graph the function and inspect the turning point.

Solve a line-parabola system

Graph both equations and inspect intersections. If linear equations themselves are shakier, our linear equations guide covers the fundamentals first.

Parameter questions

A slider can help you understand behavior, but on test day algebra may be faster for exact conditions.

For a full walkthrough of when and how to use Desmos across SAT Math, see our Desmos on the Digital SAT guide.

When not to use Desmos

If:

[ (x-7)(x+2)=0 ]

the roots are visible: 7 and -2.

Opening Desmos wastes time.

Exponential vs quadratic: don't confuse them

Quadratic: [ y=x^2 ]

Exponential: [ y=2^x ]

A quadratic has a constant second difference in equally spaced x-values.

An exponential has a constant multiplicative ratio.

The SAT may test whether you can identify the model from a table.

Context problems

Example 5

The height of an object is modeled by:

[ h(t)=-16t^2+64t+5 ]

When does it reach its maximum height?

For (at^2+bt+c), vertex x-coordinate is:

[ t=-\frac{b}{2a} ]

[ t=-\frac{64}{2(-16)}=2 ]

Maximum occurs at 2 seconds.

If the question asks only when, you do not need to calculate the maximum height.

Common SAT traps

Trap 1: losing the ±

The quadratic formula usually gives two solutions.

Trap 2: wrong sign for b

If (b=-5), then (-b=+5).

Trap 3: confusing vertex with root

Vertex is not automatically an x-intercept.

Trap 4: answering a root when asked for sum/product

Read the target carefully.

Trap 5: using the longest method

Factor when structure is obvious. Graph when graphing is obvious.

Original mini-practice

1

Solve: [ x^2-9x+20=0 ]

2

How many real solutions does [ 2x^2+4x+7=0 ] have?

A. 0
B. 1
C. 2
D. infinitely many

3

The function [ f(x)=3(x+2)^2-8 ] has what minimum value?

4

For what value of (c) does [ x^2+10x+c=0 ] have exactly one real solution?

5

A parabola has roots -1 and 6 and leading coefficient 2. Which equation represents it?

A. (y=2(x+1)(x-6))
B. (y=2(x-1)(x+6))
C. (y=(x+1)(x-6))
D. (y=2(x+1)(x+6))

Answers

4 and 5
((x-4)(x-5)=0).
A. 0
Discriminant (=4^2-4(2)(7)=16-56=-40).
-8
Vertex is ((-2,-8)), and (a>0).
25
(10^2-4c=0\Rightarrow c=25).
A
Roots -1 and 6 correspond to factors ((x+1)(x-6)).

What to learn next

exponential functions;
polynomial structure;
nonlinear systems;
function transformations.

See the rest of our SAT Math guides for other skills.

Official references

College Board, Advanced Math: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced
College Board, Student Question Bank skills: https://satsuite.collegeboard.org/practice/student-question-bank/math

Trademark note: SAT® is a registered trademark of College Board. ACT® is a registered trademark of ACT Education Corp. MastaPrep is not affiliated with, endorsed by, or sponsored by College Board or ACT Education Corp.

Ready to put this into practice?

Practice this skill free

Free to start. No credit card.

Quadratics on the Digital SAT: Factoring, Formula, Vertex & Desmos

Master Digital SAT quadratics: factoring, roots, quadratic formula, discriminant, vertex form, graph interpretation, Desmos strategy, and original practice.

Published 15 September 2026

Quadratics belong to Advanced Math, one of the two largest SAT Math domains.

Recognize the three useful forms

Standard form

[ y=ax^2+bx+c ]

Useful for:

quadratic formula;
discriminant;
y-intercept (c).

Factored form

[ y=a(x-r_1)(x-r_2) ]

Useful for:

roots / x-intercepts (r_1,r_2).

Vertex form

[ y=a(x-h)^2+k ]

Useful for:

vertex ((h,k));
minimum/maximum;
graph transformations.

A major SAT skill is choosing the form that exposes the information you need.

Factoring

Example 1

Solve:

[ x^2-7x+12=0 ]

Find two numbers that multiply to 12 and add to -7:

-3 and -4.

[ (x-3)(x-4)=0 ]

So:

[ x=3,\ 4 ]

Difference of squares

[ x^2-25=0 ]

[ (x-5)(x+5)=0 ]

[ x=\pm5 ]

Recognizing structure is faster than using the quadratic formula.

Quadratic formula

For:

[ ax^2+bx+c=0 ]

[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} ]

Example 2

[ 2x^2+x-6=0 ]

(a=2,\ b=1,\ c=-6).

[ x=\frac{-1\pm\sqrt{1-4(2)(-6)}}{4} ]

[ x=\frac{-1\pm\sqrt{49}}{4} ]

[ x=\frac{-1\pm7}{4} ]

Solutions:

[ x=\frac32,\ -2 ]

This one also factors, but the formula always gives a systematic route.

The discriminant

The expression:

[ b^2-4ac ]

tells you how many real solutions a quadratic has.

(>0): two distinct real solutions
(=0): one repeated real solution
(<0): no real solutions

Example 3

For what value of (k) does

[ x^2-6x+k=0 ]

have exactly one real solution?

Set discriminant to zero:

[ (-6)^2-4(1)(k)=0 ]

[ 36-4k=0 ]

[ k=9 ]

Vertex form

Example 4

[ f(x)=2(x-3)^2-5 ]

Vertex:

[ (3,-5) ]

Because (a=2>0), the parabola opens upward.

Minimum value:

[ -5 ]

The SAT may ask for:

minimum/maximum;
x-value at the vertex;
transformation;
a parameter.

Completing the square

Suppose:

[ x^2-8x+11 ]

Complete the square:

[ x^2-8x+16-5 ]

[ (x-4)^2-5 ]

Now the vertex is visible immediately:

[ (4,-5) ]

Intercepts

y-intercept

Set (x=0).

For:

[ y=3x^2-4x+7 ]

y-intercept = 7.

x-intercepts

Set (y=0) and solve.

These are the roots.

Desmos strategy

Desmos can make quadratics fast.

Find roots

Graph:

[ y=x^2-7x+12 ]

Click the x-intercepts.

Find vertex

Graph the function and inspect the turning point.

Solve a line-parabola system

Graph both equations and inspect intersections. If linear equations themselves are shakier, our linear equations guide covers the fundamentals first.

Parameter questions

A slider can help you understand behavior, but on test day algebra may be faster for exact conditions.

For a full walkthrough of when and how to use Desmos across SAT Math, see our Desmos on the Digital SAT guide.

When not to use Desmos

If:

[ (x-7)(x+2)=0 ]

the roots are visible: 7 and -2.

Opening Desmos wastes time.

Exponential vs quadratic: don't confuse them

Quadratic: [ y=x^2 ]

Exponential: [ y=2^x ]

A quadratic has a constant second difference in equally spaced x-values.

An exponential has a constant multiplicative ratio.

The SAT may test whether you can identify the model from a table.

Context problems

Example 5

The height of an object is modeled by:

[ h(t)=-16t^2+64t+5 ]

When does it reach its maximum height?

For (at^2+bt+c), vertex x-coordinate is:

[ t=-\frac{b}{2a} ]

[ t=-\frac{64}{2(-16)}=2 ]

Maximum occurs at 2 seconds.

If the question asks only when, you do not need to calculate the maximum height.

Common SAT traps

Trap 1: losing the ±

The quadratic formula usually gives two solutions.

Trap 2: wrong sign for b

If (b=-5), then (-b=+5).

Trap 3: confusing vertex with root

Vertex is not automatically an x-intercept.

Trap 4: answering a root when asked for sum/product

Read the target carefully.

Trap 5: using the longest method

Factor when structure is obvious. Graph when graphing is obvious.

Original mini-practice

1

Solve: [ x^2-9x+20=0 ]

2

How many real solutions does [ 2x^2+4x+7=0 ] have?

A. 0
B. 1
C. 2
D. infinitely many

3

The function [ f(x)=3(x+2)^2-8 ] has what minimum value?

4

For what value of (c) does [ x^2+10x+c=0 ] have exactly one real solution?

5

A parabola has roots -1 and 6 and leading coefficient 2. Which equation represents it?

A. (y=2(x+1)(x-6))
B. (y=2(x-1)(x+6))
C. (y=(x+1)(x-6))
D. (y=2(x+1)(x+6))

Answers

4 and 5
((x-4)(x-5)=0).
A. 0
Discriminant (=4^2-4(2)(7)=16-56=-40).
-8
Vertex is ((-2,-8)), and (a>0).
25
(10^2-4c=0\Rightarrow c=25).
A
Roots -1 and 6 correspond to factors ((x+1)(x-6)).

What to learn next

exponential functions;
polynomial structure;
nonlinear systems;
function transformations.

See the rest of our SAT Math guides for other skills.

Official references

College Board, Advanced Math: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced
College Board, Student Question Bank skills: https://satsuite.collegeboard.org/practice/student-question-bank/math

Ready to put this into practice?

Practice this skill free

Free to start. No credit card.

Recognize the three useful forms

Standard form

Factored form

Vertex form

Factoring

Example 1

Difference of squares

Quadratic formula

Example 2

The discriminant

Example 3

Vertex form

Example 4

Completing the square

Intercepts

y-intercept

x-intercepts

Desmos strategy

Find roots

Find vertex

Solve a line-parabola system

Parameter questions

When not to use Desmos

Exponential vs quadratic: don't confuse them

Context problems

Example 5

Common SAT traps

Trap 1: losing the ±

Trap 2: wrong sign for b

Trap 3: confusing vertex with root

Trap 4: answering a root when asked for sum/product

Trap 5: using the longest method

Original mini-practice

1

2

3

4

5

Answers

What to learn next

Official references

Read next

Recognize the three useful forms

Standard form

Factored form

Vertex form

Factoring

Example 1

Difference of squares

Quadratic formula

Example 2

The discriminant

Example 3

Vertex form

Example 4

Completing the square

Intercepts

y-intercept

x-intercepts

Desmos strategy

Find roots

Find vertex

Solve a line-parabola system

Parameter questions

When not to use Desmos

Exponential vs quadratic: don't confuse them

Context problems

Example 5

Common SAT traps

Trap 1: losing the ±

Trap 2: wrong sign for b

Trap 3: confusing vertex with root

Trap 4: answering a root when asked for sum/product

Trap 5: using the longest method

Original mini-practice

1

2

3

4

5