3.1A: Solving Quadratic Equations by Graphing and Factoring
Objectives: To solve a quadratic equation by graphing. To solve a quadratic equation by
factoring.
A quadratic equation is an equation that can be written in the standard form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0,
where a, b, and c are real numbers. A root of an equation is a solution of the equation.
Example: The function 𝑓(𝑥) = 𝑥 2 − 𝑥 − 6 is graphed on the right.
a) Use the graph to solve the equation 𝑥 2 − 𝑥 − 6 = 0.
b) Solve the equation 𝑥 2 − 𝑥 − 6 = 0 by factoring.
Zero Product Property: If the product of two expressions is zero, then one or both of the
expressions equal zero.
If AB = 0, then A = 0 or B = 0.
Summary: The solutions to a quadratic equation occur at the ________________________ of
the graph of the quadratic equation.
Checklist for Solving by Graphing
1. Get the equation equal to zero.
2. Graph the parabola.
3. Identify the x-intercepts.
Practice: Solve each equation by graphing.
1. −2𝑥 2 − 2 = 4𝑥
2. 𝑥 2 − 8𝑥 + 12 = 0
3. −𝑥 2 + 2𝑥 = −3
4. 2𝑥 2 + 12𝑥 + 19 = 0
Checklist for Solving by Factoring
1. Get the equation equal to zero.
2. Factor out the GCF if there is one.
3. Factor the rest further if possible.
3 terms  Reverse FOIL
2 terms  Look for difference of two squares.
Practice: Solve each equation by factoring.
5. 𝑥 2 − 4𝑥 = 45
6. 4𝑥 2 − 1 = 0
7. 3𝑥 2 − 5𝑥 = 2
Finding the Zeros of a Quadratic Function
The x-intercepts (or solutions to a quadratic equation) are also called zeros of the function
because the value of the function (the y-coordinate) is zero there.
Practice: Find the zeros of each function.
8. 𝑓(𝑥) = 2𝑥 2 − 11𝑥 + 12
9. 𝑓(𝑥) = 𝑥 2 − 8𝑥
10. 𝑓(𝑥) = 4𝑥 2 + 28𝑥 + 49
Application Problems:
11. According to legend, in 1589, the Italian scientist Galileo Galilei dropped rocks of
different weights from the top of the Leaning Tower of Pisa to prove his conjecture that the
rocks would hit the ground at the same time. The height h (in feet) of a rock after t seconds
can be modeled by h(t) = 196 – 16t2. Find and interpret the zeros of the graph.
12. Find the value of x if the area of the triangle is 42 units2.