factorization -- complete-square

Question

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

Answer 1

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

Add 121/16 to each side

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.

Answer 2

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

Add 16 to each side

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.

Answer 3

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

Add 49 to each side

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.

Answer 4

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

Add 121/144 to each side

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.

Answer 5

c).

x² + 36 = 12

Separate variables and constants aside by subtracting 36 from each side

x²  = 12 - 36

x² = - 24

(x - 0)²= - 24.