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?
minus, 3, x, squared, minus, 35, x, minus, 189, equals, 13, x
−3x
2
−35x−189=
13x
Answer
Attempt 1 out of 2

left parenthesis(
right parenthesis, squared, equals)
2
=

Answer 1

To solve the given equation by completing the square, we start with:

\[
-3x^2 - 35x - 189 = 13x
\]

First, let's rearrange the equation:

\[
-3x^2 - 35x - 13x - 189 = 0
\]

This simplifies to:

\[
-3x^2 - 48x - 189 = 0
\]

Next, we can divide the entire equation by -3 to simplify:

\[
x^2 + 16x + 63 = 0
\]

Now, let's move the constant term to the right side:

\[
x^2 + 16x = -63
\]

Now, we will complete the square for the left side. To do this, take half of the coefficient of \(x\) (which is 16), square it, and then add and subtract that value:

\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\]

So, we add 64 to both sides of the equation:

\[
x^2 + 16x + 64 = -63 + 64
\]

This gives us:

\[
x^2 + 16x + 64 = 1
\]

Now, the left side can be factored as a perfect square:

\[
(x + 8)^2 = 1
\]

Thus, the intermediate step in the form \((x + a)^2 = b\) is:

\[
(x + 8)^2 = 1
\]

In your notation, where:

- \(a = 8\) and \(b = 1\), the final answer is:

\[
\text{Answer: } (x + 8)^2 = 1
\]