The document outlines methods for solving quadratic equations by factoring, including the zero product property and various factoring strategies. It details step-by-step procedures for factoring expressions, transposing equations, and finding solutions by equating factors to zero. Examples illustrate the application of these methods to solve specific quadratic equations.

1. Solving quadratic
equations by factoring

2. Objective:
Solve quadratic equations by factoring

3. SOLVING QUADRATIC EQUATIONS BY FACTORING
Zero Product Property
Let a and b be real numbers, variables or expressions. If a and b are factors such that then .
This property applies to three or more factors as well.
Factoring Algebraic Expressions: A General Strategy
1. Factor the expression using the common monomial factor of the terms.
2. Look for special factoring patterns like Difference of Two Squares, Sum or Difference of two
cubes. Apply the method of factoring to these special algebraic expressions.
3. Find the Perfect Square Trinomials and Trinomials of the form .
4. Use factoring by grouping when necessary.

4. SOLVING QUADRATIC EQUATIONS BY FACTORING
Steps in Solving Quadratic Equations by Factoring
1. Transpose all terms on the left side of the equation if necessary.
a) Clear the equation of all fractions if necessary, then transpose.
b) Remove parentheses, then transpose.
2. Combine similar terms.
3. Factor the left side of the equation.
4. Equate each factor to zero.
5. Solve the equation in Step 4.
6. Check each root by substituting it in the original equation.

5. SOLVING QUADRATIC EQUATIONS BY FACTORING
𝒙𝟐
−𝟕 𝒙=𝟎
Solve each quadratic equation.
𝒙 (𝒙 −𝟕)=𝟎 Common Monomial Factor
𝒙=𝟎 𝒐𝒓 𝒙 − 𝟕=𝟎 Zero Property for Product
The two solutions (roots) are 0 and 7.
𝒙=𝟎 𝒐𝒓 𝒙=𝟕 Solve for x

6. SOLVING QUADRATIC EQUATIONS BY FACTORING
𝟐 𝒙𝟐
=−𝟒 𝒙
Solve each quadratic equation.
𝟐 𝒙 (𝒙+𝟐)=𝟎 Common Monomial Factor
𝟐 𝒙=𝟎𝒐𝒓 𝒙+𝟐=𝟎 Zero Property for Product
The two solutions (roots) are 0 and -2.
𝒙=𝟎 𝒐𝒓 𝒙=− 𝟐 Solve for x
𝟐 𝒙𝟐
+𝟒 𝒙=𝟎 Standard form

7. SOLVING QUADRATIC EQUATIONS BY FACTORING
Solve each quadratic equation. 𝟖 𝐱 (𝐱 −𝟏)=𝟕 𝒙𝟐
+𝟒 𝒙
𝟖 𝒙𝟐
−𝟖 𝒙=𝟕 𝒙𝟐
+𝟒 𝒙 Simplify
𝟖 𝒙𝟐
−𝟕 𝒙𝟐
−𝟖 𝒙 − 𝟒 𝒙=𝟎 Write the standard form
𝒙 𝟐
− 𝟏𝟐 𝒙 = 𝟎 Simplify
𝒙 ( 𝒙 − 𝟏𝟐 )=𝟎 Common Monomial factor
𝒙=𝟎 𝒐𝒓 𝒙 − 𝟏𝟐=𝟎 Zero Property for Product
The two solutions (roots) are 0 and 12.
𝒙=𝟎 𝒐𝒓 𝒙=𝟏𝟐 Solve for x

8. SOLVING QUADRATIC EQUATIONS BY FACTORING
Solve each quadratic equation.
Square both sides and write in standard form
𝟑 𝒙=√𝟏𝟎𝒙
𝟐
−𝟓 𝒙−𝟐𝟒
Factor and equate each factor to 0 to solve
for x.
The two solutions (roots) are and -3.

9. SOLVING QUADRATIC EQUATIONS BY FACTORING
Solve each quadratic equation. 𝑥− 2
− 2𝑥−1
−3=0
Use another variable to eliminate the negative exponent.
Let thus,
Substitute the new variable to the original equation, thus,
Factor and equate each factor to 0 to solve for x.

10. SOLVING QUADRATIC EQUATIONS BY FACTORING
Solve each quadratic equation. 𝑥− 2
− 2𝑥−1
−3=0
But, , thus, by substitution,
:
:
The two solutions (roots) are and -1.

11. SOLVING QUADRATIC EQUATIONS BY FACTORING
Solve each quadratic equation. x −8√𝑥+16=0
Use another variable to represent the radical expression.
Let thus,
Substitute the new variable to the original equation, thus,
Factor and equate each factor to 0 to solve for x.

12. SOLVING QUADRATIC EQUATIONS BY FACTORING
Solve each quadratic equation. x −8√𝑥+16=0
But, , thus, by substitution,
4
Squaring both sides to solve for x
The solution (root) is 16.

13. Activity: Find my Factors
Solve each quadratic equation. (page 15)