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Solving Equations by Factoring

WHAT TO DO: HOW TO DO IT:
1. Solve the equation x 2 + 4(x+1) 2 = (2x + 3) 2

Remove parentheses and simplify both sides:

Rewrite in standard equation form and solve the equation: ⇒

Scratch: Find a numbers whose product is 5 with difference of 4 , larger sign "−".

Set each factor equal to zero:

The solution:

x 2 + 4(x+1) 2 = (2x + 3) 2

x 2 + 4(x 2+2x+1) = 4x 2 + 12x +9

x 2 + 4x 2+8x+4 = 4x 2 + 12x +9

x 2 − 4x − 5 = 0

{5,1}

 (x − 5)(x + 1) = 0

(x − 5) = 0 or (x + 1) = 0

x = 5 or x = − 1

Check the original, equation

x 2 + 4(x+1) 2 = (2x + 3) 2

⇒ x = 5

⇒ x = -1

(5)2 + 4(5+1)2 = (2·5+3)2 , 25 + 144 = 139

(-1)2 + 4(-1+1)2 = (2·-1+3)2 , 1 + 0 = 1

2. Solve the equation x 3 = 9x 2 − 20x

Rewrite in standard equation form and solve the equation: ⇒

Factor out the common factor x ⇒

Scratch: Solve the inner equation: ⇒

Find a pair of numbers whose product is 20 with sum of 9. sign "−".

Set each factor equal to zero:

 The solution:

x 3 = 9x 2 − 20x

x 3 − 9x 2 + 20x = 0

x (x 2 − 9x + 20) = 0

x (x 2 − 9x + 20) = 0

x (x − 5)(x − 4) = 0

x = 0, (x − 5) = 0 or (x − 4) = 0

x = 0, x = 4 or x = 3

Check the original equation:

 x 3 = 9x 2 − 20x

⇒ x = 0:

⇒ x = 4:

⇒ x = 5

(0)3 = 9(0)2 − 20(0) , 0 = 0 - 0

(4)3 = 9(4)2 − 20(4) , 64 = 144 - 80

(5)3 = 9(5)2 − 20(5) , 125 = 225 - 100

 

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