1. Forms of a Quadratic Equation
x(x - 2) = 0
-4x² + 4x =1
-4x² + 4x -1 = 0
x² - 2x = 0
(x – 4)(x + 4) = 9
x² - 16 = 9
x² - 25 = 0

2. Solving EquaTIONS Using the Zero Product Rule
Definition of a Quadratic Equation in One Variable
If a, b, and c are real numbers such that a ≠ 0, then a quadratic equation is an equation that can be written in the form
ax² + bx + c = 0

3. Zero Product Rule If ab = 0, then a = 0 or b = 0 Factor
Apply the zero product rule
Set each factor equal to zero
Solve each equation for x

4. Steps for Solving a Quadratic Equation by Factoring
Write the equation in the form:
ax² + bx + c = 0
Factor the equation completely.
Apply the zero product rule, that is, set each factor equal to zero, and solve the resulting equations.
Note: The solution(s) found in Step 3 may be checked by substitution into the original equation.

5. Solve Quadratic Equation
Write the equation in the form
ax² + bx +c = 0
Factor the polynomial completely
Set each factor equal to zero
Solve each equation
Solutions

6. Solve Quadratic Equation
Write the equation in the form
ax² + bx +c = 0
Factor the polynomial completely
Set each factor equal to zero
Solve each equation
Solutions

7. Check: x = -½
Check: x = 5

8. Solve Quadratic Equation
Write the equation in the form
ax² + bx +c = 0
Factor the polynomial completely
Set each factor equal to zero
Solve each equation
Solutions

9. Solve Higher Quadratic Equation
This is a higher degree polynomial equation.
The equation is already set equal to zero. Because there are four terms, try factoring by grouping.
Solve each equation
Solutions

10. Translating to Quadratic Equation
The product of two consecutive integers is 48 more than the larger integer. Find the integers.
Let x represent the first (smaller) integer.
Then x + 1 represents the second (larger) integer
Simplify
Factor the polynomial completely
Set each factor equal to zero
Solutions
Solve each equation

11. c2 = a2 + b2 c c2 = (3)2 + (4)2 a c2 = 9 + 16 = 25 c = 5
In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse
The Pythagorean Theorem relates the lengths of the sides. c
hypotenuse
c2 = a2 + b2
leg b
c2 = (3)2 + (4)2
leg a
c2 = = 25
c = 5
If one leg of a right triangle measures 3 inches and one leg measures 4 inches, how long is the hypotenuse?

12. Applying the Pythagorean Theorem
Apply the Pythagorean theorem.
c =10
a² + b² = c² a
Substitute b= 6 and c = 10
a² + 6² = 10²
Simplify
a² + 36 = 100
a² =
b =6
a² - 64 = 0
Factor
a = a = 8
a + 8 =0 or a – 8 = 0
(a + 8)(a - 8) = 0
Set each factor equal to zero
Because x represents the length of a side of a triangle, reject the negative solution.