To write the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form, we need to follow a process called completing the square. Here is a step-by-step solution:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the terms involving [tex]\( x \)[/tex]:
[tex]\[
f(x) = 7(x^2 + 6x)
\][/tex]
2. Complete the square inside the parentheses:
- We look at the quadratic expression inside the parentheses: [tex]\( x^2 + 6x \)[/tex].
- To complete the square, we need to add and subtract a constant. This constant is found by taking half the coefficient of [tex]\( x \)[/tex] (which is 6 in this case), squaring it:
[tex]\[
\left(\frac{6}{2}\right)^2 = 3^2 = 9
\][/tex]
- Add and subtract this constant inside the parentheses:
[tex]\[
x^2 + 6x = (x^2 + 6x + 9 - 9) = (x + 3)^2 - 9
\][/tex]
3. Rewrite the quadratic function with this completed square:
[tex]\[
f(x) = 7[(x + 3)^2 - 9]
\][/tex]
4. Distribute the [tex]\( 7 \)[/tex] back into the equation:
[tex]\[
f(x) = 7(x + 3)^2 - 7 \cdot 9
\][/tex]
Simplify the constant term:
[tex]\[
7 \cdot 9 = 63
\][/tex]
So,
[tex]\[
f(x) = 7(x + 3)^2 - 63
\][/tex]
Thus, the function [tex]\( f(x) \)[/tex] written in vertex form is:
[tex]\[
f(x) = 7(x + 3)^2 - 63
\][/tex]
The correct answer is:
[tex]\[
\boxed{7(x+3)^2-63}
\][/tex]
