Certainly! Let's work through the given quadratic equation [tex]\( m(x) = x^2 + 8x - 10 \)[/tex] to rewrite it in vertex form and identify the vertex of the function.
### Step-by-Step Solution:
1. Standard Form to Vertex Form:
The standard form of a quadratic equation is [tex]\( m(x) = ax^2 + bx + c \)[/tex].
Given, [tex]\( m(x) = x^2 + 8x - 10 \)[/tex] (where [tex]\( a = 1, b = 8, \)[/tex] and [tex]\( c = -10 \)[/tex]).
We need to rewrite this in the vertex form, [tex]\( m(x) = a(x - h)^2 + k \)[/tex].
2. Complete the Square:
To rewrite the equation by completing the square:
- Identify the coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex]: Here, [tex]\( b = 8 \)[/tex].
- Find [tex]\( \left(\frac{b}{2}\right)^2 \)[/tex]:
[tex]\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16
\][/tex]
- Rewrite [tex]\( x^2 + 8x - 10 \)[/tex] to include this square:
Add and subtract 16 inside the expression while keeping the equation balanced:
[tex]\[
m(x) = x^2 + 8x - 10 = (x^2 + 8x + 16) - 16 - 10
\][/tex]
- Factor the perfect square trinomial [tex]\( (x^2 + 8x + 16) \)[/tex]:
[tex]\[
(x + 4)^2
\][/tex]
- Adjust the constant terms:
[tex]\[
(x + 4)^2 - 16 - 10 = (x + 4)^2 - 26
\][/tex]
3. Rewrite the equation in vertex form:
[tex]\[
m(x) = (x + 4)^2 - 26
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( h = -4 \)[/tex], and [tex]\( k = 6 \)[/tex].
4. Identify the vertex:
The vertex form of the equation [tex]\( m(x) = 1(x - (-4))^2 + 6 \)[/tex] reveals that the vertex of the parabola is:
[tex]\[
\text{vertex: } (-4, 6)
\][/tex]
### Summary:
- The quadratic equation in vertex form is:
[tex]\[
m(x) = (x + 4)^2 - 26
\][/tex]
- The vertex of the function is:
[tex]\[
\text{vertex: } (-4, 6)
\][/tex]
So the correct dropdown selections are [tex]\( (x + 4) \)[/tex] and [tex]\( -26 \)[/tex] for the equation, and vertex [tex]\( (-4, 6) \)[/tex].
