What is [tex][tex]$h(x) = -3x^2 - 6x + 5$[/tex][/tex] written in vertex form?

A. [tex]$h(x) = -3(x + 1)^2 + 2$[/tex]
B. [tex]$h(x) = -3(x + 1)^2 + 8$[/tex]
C. [tex][tex]$h(x) = -3(x - 3)^2 - 4$[/tex][/tex]
D. [tex]$h(x) = -3(x - 3)^2 + 32$[/tex]



Answer :

To rewrite the quadratic function [tex]\(h(x) = -3x^2 - 6x + 5\)[/tex] in vertex form, we need to complete the square. The vertex form of a quadratic function is given by [tex]\(h(x) = a(x - h)^2 + k\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Let's go through the steps to convert the given equation.

### Step 1: Factor out the coefficient of [tex]\(x^2\)[/tex]

First, we factor out the coefficient of [tex]\(x^2\)[/tex] (which is [tex]\(-3\)[/tex]) from the terms involving [tex]\(x\)[/tex]:
[tex]\[h(x) = -3(x^2 + 2x) + 5.\][/tex]

### Step 2: Complete the square

Take the coefficient of [tex]\(x\)[/tex] from inside the parentheses (which is [tex]\(2\)[/tex]), divide by 2, and square it:
[tex]\[ \left(\frac{2}{2}\right)^2 = 1. \][/tex]
Now add and subtract this square inside the parentheses:
[tex]\[ h(x) = -3(x^2 + 2x + 1 - 1) + 5. \][/tex]
This can be rewritten by grouping the perfect square trinomial and then subtracting the [tex]\(1\)[/tex] we added:
[tex]\[ h(x) = -3((x^2 + 2x + 1) - 1) + 5. \][/tex]
Simplify the group inside the parentheses:
[tex]\[ h(x) = -3((x + 1)^2 - 1) + 5. \][/tex]

### Step 3: Distribute and combine like terms

Now distribute the [tex]\(-3\)[/tex] and simplify:
[tex]\[ h(x) = -3(x + 1)^2 + 3 + 5. \][/tex]
Combine the constants:
[tex]\[ h(x) = -3(x + 1)^2 + 8. \][/tex]

So, the vertex form of the given quadratic function is:
[tex]\[ h(x) = -3(x + 1)^2 + 8. \][/tex]

Looking at the multiple-choice options, the correct answer is:
[tex]\[ h(x) = -3(x + 1)^2 + 8. \][/tex]

Therefore, the right choice is:
[tex]\[ \boxed{h(x) = -3(x+1)^2+8} \][/tex]