SAT Math skill page

SAT Quadratic Equations Practice

Handle SAT quadratic questions by recognizing roots, factors, parabolas, and the meaning of each solution.

Practice quadratic equations in Dolphin Practice questions

14-20 min practice time3 questions on pageAdvanced Math

Practice time14-20 min

Question bank3 questions

Best forAdvanced Math

What this tests

What to know for this SAT skill

Factor simple quadratic expressions
Use zeros to solve quadratic equations
Interpret roots in context
Recognize when a quadratic model has two, one, or no real solutions

Dolphin question bank

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Question 1 of 3Medium

If the given expression is rewritten in the form (2x - 3)(x + k), where k is a constant, what is the value of k?

Show answer and solution

Correct answerC. 4

Solution walkthrough

The correct answer is 4. It's given that $2 x^{2} + 5 x - 12$ can be rewritten as $(2 x - 3) (x + k)$ ; it follows that $(2 x - 3) (x + k) = 2 x^{2} + 5 x - 12$ . Expanding the left-hand side of this equation yields $2 x^{2} + 2 k x - 3 x - 3 k = 2 x^{2} + 5 x - 12$ .
Subtracting $2 x^{2}$ from both sides of this equation yields $2 k x - 3 x - 3 k = 5 x - 12$ . Using properties of equality, $2 k x - 3 x = 5 x$ and $- 3 k = - 12$ . Either equation can be solved for $k$ .
Dividing both sides of $- 3 k = - 12$ by $- 3$ yields $k = 4$ . The equation $2 k x - 3 x = 5 x$ can be rewritten as $x (2 k - 3) = 5 x$ . It follows that $2 k - 3 = 5$ .
Solving this equation for $k$ also yields $k = 4$ . Therefore, the value of $k$ is 4.

Question 2 of 3Medium

f(x) =

x^{2}

- 18x - 360 If the given function f is graphed in the xy-plane, where y = f(x), what is an x-intercept of the graph?

(-12, 0)
(-30, 0)
(-360, 0)
(12, 0)

Show answer and solution

Correct answerA. (-12, 0)

Solution walkthrough

It's given that $y = f (x)$ . The $x$ -intercepts of a graph in the $x y$ -plane are the points where $y = 0$ .
Thus, for an $x$ intercept of the graph of function $f$ , $0 = f (x)$ . Substituting $0$ for $f (x)$ in the equation $f (x) = x^{2} - 18 x - 360$ yields $0 = x^{2} - 18 x - 360$ .
Factoring the right-hand side of this equation yields $0 = (x + 12) (x - 30)$ . By the zero product property, $x + 12 = 0$ and $x - 30 = 0$ .
Subtracting 12 from both sides of the equation $x + 12 = 0$ yields $x = - 12$ . Adding 30 to both sides of the equation $x - 30 = 0$ yields $x = 30$ .
Therefore, the $x$ -intercepts of the graph of $y = f (x)$ are $(- 12, 0)$ and $(30, 0)$ . Of these two $x$ -intercepts, only $(- 12, 0)$ is given as a choice.

Question 3 of 3Medium

If the given function f is graphed in the xy-plane, where y = f(x), what is the x-coordinate of an x-intercept of the graph? f(x) = (x + 6)(x - 4)

-6
0
-4
6

Show answer and solution

Correct answerA. -6

Solution walkthrough

The correct answer is either $- 6$ or $4$ . The x-intercepts of a graph in the xy-plane are the points where $y = 0$ .
Thus, for an x-intercept of the graph of $y = f (x)$ , $0 = f (x)$ . Substituting $0$ for $f (x)$ in the equation $f (x) = (x + 6) (x - 4)$ yields $0 = (x + 6) (x - 4)$ .
By the zero product property, $x + 6 = 0$ and $x - 4 = 0$ . Subtracting $6$ from both sides of the equation $x + 6 = 0$ yields $x = - 6$ .
Adding $4$ to both sides of the equation $x - 4 = 0$ yields $x = 4$ . Therefore, the x-coordinates of the x-intercepts of the graph of $y = f (x)$ are $- 6$ and $4$ .
Note that $- 6$ and $4$ are examples of ways to enter a correct answer.

Avoid these traps

Common mistakes on this skill

Forgetting the second solution

Quadratics can have two roots. Check whether the problem wants one root, both roots, or a contextual answer.

Dropping a negative sign

Factoring and expanding quadratics often hinge on signs. Test your factors by multiplying them back out.

Using a context-impossible root

In word problems, a negative time or length may be algebraically valid but not meaningful.

Study plan

How to practice this skill in Dolphin

Memorize the factor patterns for simple trinomials.
Practice setting each factor equal to zero after factoring.
Connect roots to x-intercepts on a graph.
For word problems, reject roots that do not make sense in context.

Practice quadratic equations in Dolphin

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FAQ

Questions about SAT Quadratic Equations Practice

Do I need the quadratic formula for the SAT?

It can help, but many SAT quadratic questions are designed for factoring, graph interpretation, or simple structure recognition.

What is a root of a quadratic?

A root is an x-value that makes the quadratic equal zero. On a graph, roots are x-intercepts.

Why are quadratic word problems tricky?

They often produce more than one algebraic solution, and you have to decide which one fits the situation.