PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022
The quadratic formula
Quadratics
Passport to advanced math
LEARNING OBJECTIVE
After this lesson, you will be able to:
· Solve a quadratic equation by applying the quadratic formula
To answer a question like this:
4x2 − 7x − 4 = 0
Which of the following values of x satisfy the equation above?
A. 1 and 4
B. 
C. 
D. No real solutions
You need to know this:
The quadratic formula can be used to solve any quadratic equation. It yields solutions to a quadratic equation that is written in standard form, ax2 + bx + c = 0:
The ± sign that follows −b indicates that you will have two solutions, so remember to find both.
The expression under the radical (b2 − 4 ac) is called the discriminant, and its value determines the number of real solutions. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is equal to 0, there is only one distinct real solution. If the discriminant is negative, there are no real solutions because you cannot take the square root of a negative number.
The arithmetic can get complicated, so reserve the quadratic formula for equations that cannot be solved by factoring and those in which completing the square is difficult because a ≠ 1.
You need to do this:
Get the quadratic equation into the form ax2 + bx + c = 0. Then, substitute a, b, and c into the quadratic formula and simplify.
Explanation:
In the given equation, a = 4, b = −7, and c = −4. Plug these values into the quadratic formula and simplify:
The correct answer is (C).
Try on Your Own
Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.
HINT: For Q15, what does it mean if there are no zeros on the graph of a quadratic equation?
15. Which of the quadratic functions shown has no zeros?
A. 
B.
C.
D. 
HINT: For Q16, what does the expression under the square root sign in the quadratic formula indicate?
16. Which of the following are the real values of x that satisfy the equation 
?
A. 3 and −2
B.
C.0
D. The equation has no real solutions.
17. Which of the following are the roots of the equation x2 + 8x − 3 = 0?
A. 
B.
C.
D. 
HINT: For Q18, remember that for a parabola to have two real solutions, the discriminant b2 − 4 ac must be greater than zero.
18. Which quadratic equation has two real solutions?
A. 
B.
C.
D. 
19. What are the roots of the above equation?
A. 
B.
C.
D. 