Question
You notice that the following quadratic equation cannot be factored, so you solve it by completing the square. What value of c would make the left-hand side of this equation a perfect square trinomial?
x^ 2 - 8x + c = 13
(1 point)
4
64
16
-8
-4
x^ 2 - 8x + c = 13
(1 point)
4
64
16
-8
-4
Answers
GPT-4o mini
To complete the square for the quadratic equation \( x^2 - 8x + c = 13 \), we need to determine the value of \( c \) that makes the left-hand side a perfect square trinomial.
The general form for completing the square of the expression \( x^2 + bx \) is:
\[
x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2
\]
In this case, our \( b \) is \(-8\):
1. Calculate \( \frac{b}{2} \):
\[
\frac{-8}{2} = -4
\]
2. Calculate \( \left(\frac{b}{2}\right)^2 \):
\[
(-4)^2 = 16
\]
This means that in order to make \( x^2 - 8x + c \) a perfect square trinomial, we need:
\[
c = 16
\]
Thus, the expression \( x^2 - 8x + 16 \) can be written as \( (x - 4)^2 \).
Therefore, the value of \( c \) that would make the left-hand side a perfect square trinomial is \( \boxed{16} \).
The general form for completing the square of the expression \( x^2 + bx \) is:
\[
x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2
\]
In this case, our \( b \) is \(-8\):
1. Calculate \( \frac{b}{2} \):
\[
\frac{-8}{2} = -4
\]
2. Calculate \( \left(\frac{b}{2}\right)^2 \):
\[
(-4)^2 = 16
\]
This means that in order to make \( x^2 - 8x + c \) a perfect square trinomial, we need:
\[
c = 16
\]
Thus, the expression \( x^2 - 8x + 16 \) can be written as \( (x - 4)^2 \).
Therefore, the value of \( c \) that would make the left-hand side a perfect square trinomial is \( \boxed{16} \).