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Solving Quadratic Equations by Factoring
Jul 23, 2024
Solving Quadratic Equations by Factoring
Overview
Learn how to solve quadratic equations by factoring
Tutorial divided into two parts
Part 1
: Two-term quadratics (using Greatest Common Factor, Difference of Squares)
Part 2
: Trinomials (leading coefficient of one or different from one)
Also covers non-standard form and common mistakes
Part 1: Solving Two-Term Quadratic Equations
Greatest Common Factor (GCF)
Example 1:
Equation: x^2 - 10x = 0
GCF of x^2 and -10x is x
Factored form: x(x - 10)
Zero Product Property: x = 0 or x - 10 = 0
Solutions: x = 0 or x = 10
Example 2:
Equation: 3x^2 + 15x = 0
GCF of 3x^2 and 15x is 3x
Factored form: 3x(x + 5)
Solutions: x = 0 or x = -5
Difference of Two Squares
Example 3:
Equation: x^2 - 81 = 0
81 is a perfect square: 9^2
Factored form: (x + 9)(x - 9)
Solutions: x = -9 or x = 9
Example 4:
Equation with negative leading coefficient: -x^2 + 49 = 0
Factor out -1: -(x^2 - 49) = 0
Factored form: - (x + 7)(x - 7)
Solutions: x = -7 or x = 7
Part 2: Solving Quadratic Trinomials
Leading Coefficient of One
Example 5: Product-Sum Method
Equation: x^2 + 11x + 24 = 0
Product of 24, sum of 11: Factor pairs (3, 8)
Factored form: (x + 3)(x + 8)
Solutions: x = -3 or x = 8
Example 6: Negative Product
Equation: x^2 - 5x - 36 = 0
Product of -36, sum of -5: Factor pairs (4, -9)
Factored form: (x + 4)(x - 9)
Solutions: x = -4 or x = 9
Leading Coefficient Not Equal to One
Example 7:
Equation: -x^2 + 42x + 63 = 0
Factor out -1: -(x^2 - 42x - 63) = 0
Product-Sum Method: Factors (3, 14)
Factored form: -(x - 3)(x + 14)
Solutions: x = 3 or x = -14
Example 8: AC Method
Equation: 5x^2 + 18x + 9 = 0
Multiply leading coefficient by constant: 5 * 9 = 45
Factors of 45 that add to 18: (3, 15)
Split middle term, factor by grouping:
Group 1: 5x^2 + 15x (GCF = 5x)
Group 2: 3x + 9 (GCF = 3)
Factored form: (5x + 3)(x + 3)
Solutions: x = -3/5 or x = -3
Non-Standard Form
Example 9:
Initial Equation: 4x(x - 5) = -25
Distribute and rearrange: 4x^2 - 20x + 25 = 0
Use AC Method:
Product is 100, sum is -20
Factors: (-10, -10)
Factored form: (2x - 5)^2 = 0
Solution: x = 5/2
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