Question

What is the intermediate step in the form left parenthesis, x, plus, a, right parenthesis, squared, equals, b(x+a)
2
=b as a result of completing the square for the following equation?
x, squared, plus, 28, equals, minus, 8, x
x
2
+28=
−8x

Answer 1

To solve the equation \( x^2 + 28 = -8x \) using the method of completing the square, we can first rearrange it into a more standard form. Here are the steps:

1. Start with the original equation:
\[
x^2 + 28 = -8x
\]
We can rearrange it:
\[
x^2 + 8x + 28 = 0
\]

2. Next, we need to isolate the quadratic and linear terms on one side. To do this, we'll move the constant term to the other side:
\[
x^2 + 8x = -28
\]

3. Now, we will complete the square. To do this, take half of the coefficient of \( x \) (which is \( 8 \)), square it, and add it to both sides. Half of \( 8 \) is \( 4 \), and squaring it gives \( 16 \):
\[
x^2 + 8x + 16 = -28 + 16
\]
Simplifying the right side:
\[
x^2 + 8x + 16 = -12
\]

4. The left side can be factored as a perfect square:
\[
(x + 4)^2 = -12
\]

Now we have completed the square, and we obtain the intermediate step:
\[
(x + 4)^2 = -12
\]

This is the intermediate step in the form \( (x + a)^2 = b \) where \( a = 4 \) and \( b = -12 \).