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, 6, x, squared, plus, 64, x, minus, 222, equals, minus, 8, x
−6x
2
+64x−222=
−8x

Answer 1

To solve the equation \(-6x^2 + 64x - 222 = -8x\) by completing the square, we start by rearranging the equation into a standard quadratic form. Let's first move \(-8x\) to the left side:

\[
-6x^2 + 64x + 8x - 222 = 0
\]

This simplifies to:

\[
-6x^2 + 72x - 222 = 0
\]

Next, we can simplify this equation by dividing everything by \(-6\):

\[
x^2 - 12x + 37 = 0
\]

Now, we will complete the square on the quadratic \(x^2 - 12x + 37\). The coefficient of \(x\) is \(-12\). To complete the square:

1. Take half of \(-12\), which is \(-6\), and square it to get \(36\).
2. Rewrite the equation as:

\[
(x - 6)^2 - 36 + 37 = 0
\]
\[
(x - 6)^2 + 1 = 0
\]

At this point, we have completed the square. If we were to set the completed square form equal to a constant:

\[
(x - 6)^2 = -1
\]

Thus, the intermediate step of the form \((x + a)^2 = b\) involves noting that in our case:

\((x - 6)^2 = -1\)

Here \(a = -6\) and \(b = -1\).

So, the intermediate step after completing the square is:

\[
(x - 6)^2 = -1
\]

If needed in the requested form, it would be noted as:

\[
(x + (-6))^2 = -1
\]

This is the required form after completing the square for the given equation.