To rewrite the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form, we need to complete the square.
The vertex form of a quadratic function is:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
1. Starting with the given function:
[tex]\[ f(x) = 7x^2 + 42x \][/tex]
2. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[ f(x) = 7(x^2 + 6x) \][/tex]
3. To complete the square inside the parentheses, we need to add and subtract the square of half the coefficient of [tex]\( x \)[/tex]:
- The coefficient of [tex]\( x \)[/tex] is 6.
- Half of 6 is 3.
- The square of 3 is 9.
So, we add and subtract 9 inside the parentheses:
[tex]\[ f(x) = 7(x^2 + 6x + 9 - 9) \][/tex]
[tex]\[ f(x) = 7((x^2 + 6x + 9) - 9) \][/tex]
[tex]\[ f(x) = 7((x + 3)^2 - 9) \][/tex]
4. Distribute the 7 to both terms inside the parentheses:
[tex]\[ f(x) = 7(x + 3)^2 - 7 \cdot 9 \][/tex]
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
So the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Therefore, the correct answer is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
