Math 2 Honors
Lesson 3-7: Solving Quadratic Equations with the
Quadratic Formula
Name ________________________
Date _________________________
Learning Goals





I can solve quadratic equations with real numbers as coefficients by inspection (graphing), by
using the quadratic formula.
I can determine the number of solutions for a quadratic equation in standard form, ax2+bx+c =
0, by calculating the discriminant.
I can explain that complex solutions result when the radicand is negative in the quadratic
formula (b2-4ac<0).
I can determine when a quadratic equation in standard form, ax2+bx+c=0, has complex roots by
looking at a graph of f(x) = ax2+bx+c or by calculating the discriminant.
I can write complex number solutions for a quadratic equation in the form a+bi by using i=√−1.
Derivation of the Quadratic Formula
f ( x )  ax 2  bx  c
OVER 
When we cannot factor to solve a quadratic equation (or the equation is difficult to factor), we can use
the quadratic formula. For any quadratic equation in the form ax 2  bx  c  0 , the quadratic formula
allows us to solve for x without factoring or complete the square. The quadratic formula is:
b
b2  4ac
x

2a
2a
1.
or
b  b2  4ac
x
2a
Use the quadratic formula to solve the equations below. Be sure to also graph the equations in
your calculator in order to check your solutions.
a.
x2 + 6x – 9 = 0
Quick Sketch of Graph:
b.
2x2 – 12x + 18 = 0
Quick Sketch of Graph:
c.
0 = 20 – 6x + 3x2
Quick Sketch of Graph:
In problem number 1, we discovered that when finding the solutions of a quadratic function there were
different types of solutions.
2.
List the different types of possible solutions that you discovered for a quadratic function:
A way to find out the type of solution(s) of quadratic function before using the quadratic formula is to
calculate the discriminant ( b 2  4ac ).
3.
a.
Calculate the discriminant for problem 1a.
b.
If the discriminant is ___________________ (positive, negative, or zero), then there are
____________ solution(s).
4.
a.
Calculate the discriminant for problem 1b.
b.
If the discriminant is ___________________ (positive, negative, or zero), then there are
____________ solution(s).
5.
a.
Calculate the discriminant for problem 1c.
b.
If the discriminant is ___________________ (positive, negative, or zero), then there are
____________ solution(s).
OVER 
6.
For a quadratic function with the rule in the form f ( x)  ax 2  bx  c :
a. What information about the graph is provided by
b
b2  4ac
b
b2  4ac

and

?
2a
2a
2a
2a
b. What information about the graph is provided by the expression
c. What information about the graph is provided by the expression
b
?
2a
b2  4ac
?
2a