# factorization -- complete-square

a) x^2+4x=12x comprobación

b) x^2+24x=10x

c) x^2+36=12

d) 2x^2=11x-15

e) 6x^2+11x+3=0

d).

2x2 = 11x - 15

Separate variables and constants aside by subtracting 11x from each side

2x2 - 11x = - 15

Devide each side 2.

x2 - (11/2)x = - 15/2

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression

Here x coefficient = - 11/2. so, (half the x coefficient)² = (- 11/4)2= 121/16

x2 - (11/2)x + (121/16) = (- 15/2) + (121/16)

(x - 11/4)2= (- 120 + 121)/16

(x - 11/4)2= 1/16

(x - 11/4)2 - 1/16 = 0

(x - 11/4)2 - (1/4)2 = 0

Use difference of two squares formula : a2 - b2 = (a + b)(a - b).

(x - 11/4 + 1/4)(x - 11/4 - 1/4) = 0

(x - 10/4)(x - 12/4) = 0

(x - 5/2)(x - 3) = 0.

a).

x2 + 4x = 12x

Separate variables and constants aside by subtracting 12x from each side

x2 + 4x - 12x = 0

x2 - 8x = 0

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression

Here x coefficient = - 8. so, (half the x coefficient)² = (- 8/2)2= 16

x2 - 8x + 16 = 0 + 16

(x - 4)2= 16

Subtract 16 from each side.

(x - 4)2 - 16 = 0

(x - 4)2 - 42 = 0

Use difference of two squares formula : a2 - b2 = (a + b)(a - b).

(x - 4 + 4)(x - 4 - 4) = 0

x(x - 8) = 0.

b).

x2 + 24x = 10x

Separate variables and constants aside by subtracting 10x from each side

x2 + 24x - 10x = 0

x2 + 14x = 0

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression

Here x coefficient = 14. so, (half the x coefficient)² = (14/2)2= 49

x2 + 14x + 49 = 0 + 49

(x + 7)2= 49

Subtract 49 from each side.

(x + 7)2 - 49 = 0

(x + 7)2 - 72 = 0

Use difference of two squares formula : a2 - b2 = (a + b)(a - b).

(x + 7 + 7)(x + 7 - 7) = 0

(x + 14)x = 0.

e).

6x2 + 11x + 3 = 0.

Separate variables and constants aside by subtracting 11x from each side

6x2 + 11x = - 3

Devide each side 6.

x2 + (11/6)x = - 3/6

x2 + (11/6)x = - 1/2

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression

Here x coefficient = 11/6. so, (half the x coefficient)² = (11/12)2= 121/144

x2 + (11/6)x + (121/144) = (- 1/2) + (121/144)

(x + 11/12)2 = (- 72 + 121)/144

(x + 11/12)2 = 49/144

(x + 11/12)2 - 49/144 = 0

(x + 11/12)2 - (7/12)2 = 0

Use difference of two squares formula : a2 - b2 = (a + b)(a - b).

(x + 11/12 + 7/12)(x + 11/12 - 7/12) = 0

(x + 18/12)(x + 4/12) = 0

(x + 6/2)(x + 1/3) = 0.

c).

x2 + 36 = 12

Separate variables and constants aside by subtracting 36 from each side

x2  = 12 - 36

x2 = - 24

(x - 0)2= - 24.