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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, minus, 4, x, plus, 61, equals, 10, x, plus, 13
x
2
−4x+61=
10x+13
Answer
Attempt 1 out of 5

Answer 1

To solve the equation \( x^2 - 4x + 61 = 10x + 13 \) by completing the square, we first rearrange the equation so that all terms are on one side:

\[
x^2 - 4x + 61 - 10x - 13 = 0
\]

This simplifies to:

\[
x^2 - 14x + 48 = 0
\]

Next, we want to complete the square for the expression \( x^2 - 14x \). We take the coefficient of \( x \) (which is \(-14\)), halve it to get \(-7\), and then square it to get \(49\).

Now, we can rewrite the equation by adding and subtracting \(49\):

\[
x^2 - 14x + 49 - 49 + 48 = 0
\]

This simplifies to:

\[
(x - 7)^2 - 1 = 0
\]

Now, let's transform this into the form \( (x + a)^2 = b \):

Adding \(1\) to both sides gives:

\[
(x - 7)^2 = 1
\]

Thus, we have:

\[
(x - 7)^2 = 1
\]

In the form \( (x + a)^2 = b \), \( a \) would be \(-7\) (since \( x - 7 = x + (-7) \)) and \( b = 1 \).

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

\[
(x - 7)^2 = 1
\]