Pleaseee help!!

Question

Show how to complete the square for each quadratic function. The form y= ax^2 + bx + c form to y= a( x- p)^2 + q form.

a) Y= 2x^2 + 24x + 7

b) Y= -3x^2 + 12x + 7

c) Y= 3x^2 + 8x + 10

Thank You so much!

Answer 1

a) The function y = 2x² + 24x + 7

Standard form of parabola y  = ax ² + bx + c

Vertex form of parabola is y  = a (x - h )² + k

Here x² coefficient is 2 , for perfect square make x² coefficient 1 by dividing each side by 2.

y/2 = (2x² + 24x + 7)/2

y/2 = x² + 12x + 7/2

To complete the square add (half coefficient of x) ² to each side of the expression.

x coefficient = 12 then (half the x coefficient)² is (6)² = 36.

y/2 + 36 = x² + 12x + 7/2 + 36

y/2 = (x² + 12x + 36) + 7/2 - 36

y/2 = (x + 6)² - 65/2

y = 2 (x + 6)² - (65/2)2

y = 2 (x + 6)² - 65

y = 2 [ x - (-6)]² - 65

Now which is in y  = a (x - p)² + q form.

Answer 2

b) The function y = -3x² +12x + 7

Standard form of parabola y  = ax ² + bx + c

Vertex form of parabola is y  = a (x - h )² + k

Here x² coefficient is -3 , for perfect square make x² coefficient 1 by dividing each side by negative 3.

y/(-3) = (-3x² +12x + 7)/(-3)

y/(-3) = x² - 4x - (7/3)

To complete the square add (half coefficient of x) ² to each side of the expression.

x coefficient = - 4 then (half the x coefficient)² is (-2)² = 4.

y/(-3) + 4 = x² - 4x - (7/3) + 4

y/(-3) = x² - 4x + 4 - (7/3) - 4

y/(-3) = (x - 2)² - 19/3

y = (-3) (x - 2)² - (19/3)(-3)

y = (-3) (x - 2)² + 19

Now which is in vertex  form y  = a (x - p)² + q.

Answer 3

c) The function y = 3x² + 8x + 10

Standard form of parabola y  = ax ² + bx + c

Vertex form of parabola is y  = a (x - h )² + k

Here x² coefficient is 3 , for perfect square make x² coefficient 1 by dividing each side by 3.

y/3 = (3x² + 8x + 10)/3

y/3= x² + (8/3)x + (10/3)

To complete the square add (half coefficient of x) ² to each side of the expression.

x coefficient = 8/3 then (half the x coefficient)² is (4/3)² =16/9.

y/3 + (16/9) = x² + (8/3)x + (10/3) + (16/9)

y/3  = x² + (8/3)x + (16/9) + (10/3) - (16/9)

y/3  = [ x + (4/3)]² + 14/9

y = (3) [ x + (4/3)]² + (14/9)(3)

y = (3) [ x - (-4/3)]² + (14/3)

Now which is in y = a (x - p)² + q form.