1. Solving Quadratic Equations Section 1.3

2. What is a Quadratic Equation?
A quadratic equation in x is an equation that can be written in the standard form:
ax² + bx + c = 0
Where a,b,and c are real numbers and
a ≠ 0.

3. Solving a Quadratic Equation by Factoring.
The factoring method applies the zero product property:
Words: If a product is zero, then at least one of its
factors has to be zero.
Math: If (B)(C)=0, then B=0 or C=0 or both.

4. Recap of steps for how to solve by Factoring
Set equal to 0
Factor
Set each factor equal to 0 (keep the squared term positive)
Solve each equation (be careful when determining solutions, some may be imaginary numbers)

5. Example 1 Solve x² - 12x + 35 = 0 by factoring.
Set each factor equal to zero by the zero product property.
Solve each equation to find solutions.
The solution set is:
(x – 7)(x - 5) = 0
(x – 7)=0 (x – 5)=0
x = 7 or x = 5
{ 5, 7 }

6. Example 2 Solve 3t² + 10t + 6 = -2 by factoring.
Check equation to make sure it is in standard form before solving. Is it?
It is not, so set equation equal to zero first:
3t² + 10t + 8 = 0
Now factor and solve.
(3t + 4)(t + 2) = 0
3t + 4 = t +2 = 0
t = t = -2

7. Solve by factoring.

8. Solve by the Square Root Method.
If the quadratic has the form ax² + c = 0, where a ≠ 0, then we could use the square root method to solve.
Words: If an expression squared is equal to a constant, then that expression is equal to the positive or negative square root of the constant.
Math: If x² = c, then x = ±c.
Note: The variable squared must be isolated first (coefficient equal to 1).

9. Example 1: Solve by the Square Root Method:=
x = ± 4

10. Example 2: Solve by the Square Root Method.
x = ±i

11. Example 3: Solve by the Square Root Method.
x – 3 = 5 or x – 3 = -5
x = x = -2

12. Solve by the Square Root Method

13. Solve by Completing the Square.
Words
Express the quadratic equation in the following form.
Divide b by2 and square the result, then add the square to both sides.
Write the left side of the equation as a perfect square.
Solve by using the square root method.
Math
x² + bx = c
x² + bx + ( )² = c + ( )²
(x )² = c + ( )²

14. Example 1: Solve by Completing the Square.
x² + 8x – 3 = 0
x² + 8x = 3
x² + 8x + (4)² = 3 + (4)²
x² + 8x + 16 =
(x + 4)² = 19
x + 4 = ±
x = -4 ±
Add three to both sides.
Add ( )² which is (4)² to both sides.
Write the left side as a perfect square and simplify the right side.
Apply the square root method to solve.
Subtract 4 from both sides to get your two solutions.

15. Example 2: Solve by Completing the Square when the Leading Coefficient is not equal to 1.
2x² - 4x + 3 = 0
x² - 2x = 0
x² - 2x + ___ = ____
x² - 2x + 1 =
(x – 1)² =
x – 1 = ±
x = 1 ±
Divide by the leading coefficient.
Continue to solve using the completing the square method.
Simplify radical.

16. Quadratic Formula
If a quadratic can’t be factored, you must use the quadratic formula.
If ax² + bx + c = 0, then the solution is:

17. a = 1
b = -4
c = -1
Solve

18. Solve

19. Solve

20. Discriminant
The term inside the radical b² - 4ac is called the discriminant.
The discriminant gives important information about the corresponding solutions or answers of ax² + bx + c = 0, where a,b, and c are real numbers.
b² - 4ac
Solutions
b² - 4ac > 0
b² - 4ac = 0
b² - 4ac < 0

21. Tell what kind of solution to expect