1. Solving Quadratic Equationsx y -2 -1 1 3 4

2. Quadratic Equation y = ax2 + bx + c
We will learn how to solve a quadratic equation by…
Taking the square root
Factoring and then solving
Examining a graph or table
Using the Quadratic Formula

3. Square Root Method When the equation is ax2 + c = 0:
Isolate the x2, and take the square root of both sides to cancel the exponent
Example:
Key points:
Use ± to show that x was either positive or negative before being squared. Simplify the square root when necessary.

4. Factor and Solve, when a=1
From Math 1, you factored trinomials:
x2 + 4x – 12  (x + 6)(x – 2)
When solving a similar trinomial equation, set each ( ) equal to 0 to solve for x:
x2 + 4x – 12 = 0
(x + 6)(x – 2) = 0
x + 6 = 0 or x – 2 = 0
x = -6 or x = 2

5. Factor and Solve, when a > 1
In Math 2 you learned how to factor if there is a coefficient of the squared term:
2x2 – 7x – 4  (2x + 1)(x – 4)
Solve an equation by setting each ( ) equal to 0 again, but expect some fractions!
2x2 – 7x – 4 = 0
(2x + 1)(x – 4) = 0
2x + 1 = 0 or x – 4 = 0
x = -1/2 or x = 4

6. Solving Quadratic Equations with Graphs and Tables
Since the number of solutions is the same as the number of x-intercepts, the number of real solutions
is at most two.
No solutions
One solution
Two solutions

7. Solving Equations
When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts.
These values are also referred to as solutions, zeros, or roots.

8. Solutions to Quadratic Equations
x-intercepts
These terms all mean the same thing.
solutions
roots
zeros

9. Identifying Solutions
Example f(x) = x2 - 4
Solutions are -2 and 2.

10. Identifying Solutions
Now you try this problem.
What are the solutions to f(x) = 2x - x2 ?
Solutions are 0 and 2.

11. Identifying Solutions from Tables
One method of graphing uses a table with x-values.
Look at the graph of y = x2 - 4x
Roots 0 and 4
Look at the table of values that
goes with the graph.
What do you notice
about the y value for
the roots? x y 1 -3 2 -4 3 4

12. Graphing Quadratic Equations
Look at the parabola y = x2 - 2x - 8.
What are the solutions?
What would the table look
like at those values? x y -2 -1 -5 1 -9 3 4

13. The Quadratic Formula (from Math 2)
Yes! The Quadratic Formula is provided on your GHSGT formula sheet!
Where a, b, and c are the numbers (coefficients) from the equation:
ax2 + bx + c = 0

14. Example
3x2 – 8x + 4 = 0
x = 2 or x = 2/3

15. Complex Numbers
Recall from Math 2 that i=√-1, so that anytime you have a negative under the square root, simplify with i.
Example: √-64 = 8i
Example: √-120  Factor Tree!
= 2i√30

16. Complex Solutions
Solve using Quadratic Formula:
x2 – 5x + 9 = 0