Question

Solve the quadratic equation by completing the square. What are the missing pieces to the steps?

-27 = 4x²- 24x
-27 = 4(x²- 6x + 9)
9 =4(x²- 6x + 9)
9/4 = (x -3)
+__= x - 3
__ = x​

Answer 1

I think the missing steps are:

`(3)/(2)=x-3`

and

`(9)/(2)=x`

Explanation:

`-27=4x^2-24x`

First, factor 4 to make isolate
`x^2`

`-27=4(x^2-6x)`

Now, divide by 4.

`-(27)/(4)=x^2-6x`

Complete the square by adding
`((b)/(2))^2`

`((b)/(2))^2-(27)/(4)=x^2-6x+ ((b)/(2))^2`

`((6)/(2))^2-(27)/(4)=x^2-6x+ ((6)/(2))^2`

`(3)^2-(27)/(4)=x^2-6x+ (3)^2`

`9-(27)/(4)=x^2-6x+9`

Solve the difference.

`(9*4-27)/(4)=x^2-6x+9`

`(36-27)/(4)=x^2-6x+9 \\(9)/(4)=x^2-6x+9`

Factor the equation.

`(9)/(4)=(x-3)(x-3)`

Rewrite it;

`(9)/(4)=(x-3)^2`

Extract the square root.

`\sqrt{(9)/(4) }=x-3`

Take the square root of 9 and 4.

`(3)/(2)=x-3`

Add 3.

`(3)/(2)+3=x`

Do the sum

`(3+2*3)/(2)=x\\(3+6)/(2)=x\\(9)/(2)=x`

Answer 2

The missing pieces should be after line 3.

9/4 = (x - 3)^2

<=> (3/2)^2 = (x - 3)^2

<=> x - 3 = 3/2 or x - 3 = -3/2

<=> x = 3 + 3/2 or x = 3 - 3/2

<=> x = 9/2 or x = 3/2

Hope this helps!

:)