Learn powerful factoring techniques to solve quadratic equations efficiently. Discover common factors, difference of squares, and trinomial factoring methods.
A quadratic equation in standard form is: ax² + bx + c = 0 (where a ≠ 0)
• a = coefficient of x² (quadratic coefficient)
• b = coefficient of x (linear coefficient)
• c = constant term
Step 1: Look for common factors
Step 2: Check for difference of squares (a² - b²)
Step 3: Factor trinomials (ax² + bx + c)
Step 4: Set each factor equal to zero
Solve: 2x² - 8x = 0
Step 1: Factor out the common factor 2x
Step 2: Set each factor equal to zero
2x = 0
x = 0
x - 4 = 0
x = 4
Solutions: x = 0 or x = 4
Solve: 3x² + 12x + 9 = 0
Step 1: Factor out the common factor 3
Step 2: Factor the trinomial inside parentheses
Step 3: Complete factorization
Step 4: Solve each factor
x + 1 = 0
x = -1
x + 3 = 0
x = -3
Solutions: x = -1 or x = -3
Difference of squares: a² - b² = (a + b)(a - b)
Solve: x² - 16 = 0
Step 1: Recognize the pattern (perfect squares)
Step 2: Apply the difference of squares formula
Step 3: Set each factor equal to zero
x + 4 = 0
x = -4
x - 4 = 0
x = 4
Solutions: x = -4 or x = 4
Solve: 9x² - 25 = 0
Step 1: Identify perfect squares
Step 2: Apply the formula
Step 3: Solve each factor
3x + 5 = 0
x = -5/3
3x - 5 = 0
x = 5/3
Solutions: x = -5/3 or x = 5/3
Step 1: Find two numbers that multiply to ac and add to b
Step 2: Rewrite the middle term using these numbers
Step 3: Factor by grouping
Solve: x² + 7x + 12 = 0
Step 1: Find two numbers that multiply to 12 and add to 7
Step 2: Factor the trinomial
Step 3: Set each factor equal to zero
x + 3 = 0
x = -3
x + 4 = 0
x = -4
Solutions: x = -3 or x = -4
Solve: 2x² + 11x + 5 = 0
Step 1: Find two numbers that multiply to 2×5=10 and add to 11
Step 2: Rewrite the middle term
Step 3: Factor by grouping
Step 4: Solve each factor
x + 5 = 0
x = -5
2x + 1 = 0
x = -1/2
Solutions: x = -5 or x = -1/2
After factoring (x + 3)(x - 2), don't forget to solve x + 3 = 0 and x - 2 = 0
When solving x - 4 = 0, remember to add 4 to both sides: x = 4 (not x = -4)
Always look for common factors before attempting other factoring methods
Example: 2x² + 6x + 3x + 9
= 2x(x + 3) + 3(x + 3)
= (2x + 3)(x + 3)
When to use: Four terms with common factors in pairs
Pattern: a² ± 2ab + b²
= (a ± b)²
Example: x² + 6x + 9 = (x + 3)²
Projectile Motion
h = -16t² + vt + h₀
Finding when object hits ground (h = 0)
Profit Optimization
P = -x² + 100x - 500
Finding break-even points (P = 0)
Area Calculations
A = x(20 - x)
Optimizing rectangular areas
Problem 1:
Solve: 4x² - 36 = 0
4x² - 36 = 4(x² - 9) = 4(x + 3)(x - 3) = 0
Solutions: x = -3 or x = 3
Problem 2:
Solve: x² - 5x + 6 = 0
x² - 5x + 6 = (x - 2)(x - 3) = 0
Solutions: x = 2 or x = 3
Problem 3:
Solve: 3x² - 12x = 0
3x² - 12x = 3x(x - 4) = 0
Solutions: x = 0 or x = 4