To rewrite the quadratic equation in vertex form and identify the vertex, follow these detailed steps:
1. Given Equation:
[tex]\[
m(x) = x^2 + 8x - 10
\][/tex]
2. Rewrite in Vertex Form:
We need to complete the square to rewrite the equation in the form [tex]\( m(x) = a(x - h)^2 + k \)[/tex].
- Start with the quadratic and linear terms:
[tex]\[
x^2 + 8x
\][/tex]
- To complete the square, take half of the coefficient of [tex]\( x \)[/tex], square it, and add and subtract it:
[tex]\[
x^2 + 8x + 16 - 16
\][/tex]
- This can be grouped as a perfect square trinomial:
[tex]\[
(x + 4)^2 - 16
\][/tex]
- Now, include the constant term from the original equation:
[tex]\[
(x + 4)^2 - 16 - 10 = (x + 4)^2 - 26
\][/tex]
3. Vertex Form:
The vertex form of the equation is:
[tex]\[
m(x) = (x + 4)^2 - 26
\][/tex]
Thus, filling in the blanks:
[tex]\[
m(x) = (x + 4 )^2 - 26
\][/tex]
4. Identify the Vertex:
From the vertex form [tex]\( m(x) = (x + 4)^2 - 26 \)[/tex], we can see that the vertex [tex]\((h, k)\)[/tex] is determined by the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] in the equation [tex]\( a(x - h)^2 + k \)[/tex].
Here, [tex]\( h = -4 \)[/tex] and [tex]\( k = -26 \)[/tex]. So, the vertex is:
[tex]\[
(-4, -26)
\][/tex]
Thus, the correct completed statements are:
[tex]\[
m(x) = (x + 4)^2 - 26
\][/tex]
[tex]\[
\text{vertex: } (-4, -26)
\][/tex]
