help please!

Question

show how to complete the square for each quadratic function.change 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

Answer 1

(a).

Completing the square Method :

The expression is y = 2x² + 24x + 7.

Take out common factor.

y = 2(x² + 12x + 7/2)

To change the expression (x² + 12x) into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

 Here x coefficient = 12. So, (half the x coefficient)² = (12/2)² = 36.

Add and subtract 36 to the expression.

y = 2(x² + 12x + 7/2 + 36 - 36)

y = 2(x² + 12x + 36 + 7/2 - 36)

y = 2(x² + 12x + 36 - 65/2)

y = 2(x² + 2(6)(x) + 6²) - 65

Apply Perfect Square Trinomial : u² + 2uv + v² = (u + v)².

y = 2(x + 6)² - 65

The solution is y = 2[x - (-6)]² + (-65).

Answer 2

(b).

Completing the square Method :

The expression is y = -3x² + 12x + 7.

Take out common factor.

y = -3(x² - 4x - 7/3)

To change the expression (x² - 4x) into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

 Here x coefficient = 4. So, (half the x coefficient)² = (4/2)² = 4.

Add and subtract 4 to the expression.

y = -3(x² - 4x + 4 - 4 - 7/3)

y = -3(x² - 4x + 4 - 19/3)

y = -3(x² - 4x + 2²) + 19

Apply Perfect Square Trinomial : u² - 2uv + v² = (u - v)².

y = -3(x - 2)² + 19

The solution is y = -3(x - 2)² + 19.

Answer 3

(c).

Completing the square Method :

The expression is y = 3x² + 8x + 10.

Take out common factor.

y = 3(x² + 8x/3 + 10/3)

To change the expression (x² + 8x/3) into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

 Here x coefficient = 8/3. So, (half the x coefficient)² = (8/6)² = 16/9.

Add and subtract 16/9 to the expression.

y = 3(x² + 8x/3 + 16/9 -16/9 + 10/3)

y = 3(x² + 8x/3 + 16/9 + 14/9)

y = 3(x² + 2(4/3)(x) + (4/3)²] + 14/3

Apply Perfect Square Trinomial : u² + 2uv + v² = (u + v)².

y = 3(x + 4/3)² + 14/3

The solution is y = 3[x - (-4/3)]² + 14/3.