Write the equation of a quadratic function with the given solutions. Use standard form $$ f(x)=a x^{2}+b x+c $$ $$ x=11 $$ and $$ x=-4 $$

Question

UpStudy AI · Accepted Answer

The quadratic function with solutions $$ x=11 $$ and $$ x=-4 $$ can be written in factored form as

$$
f(x) = a(x - 11)(x + 4)
$$

where $$ a $$ is a nonzero constant. To write it in the standard form $$ f(x) = ax^2 + bx + c $$, we expand the factors.

First, expand the product:

$$
\begin{aligned}
(x - 11)(x + 4) &= x(x + 4) - 11(x + 4) \
&= x^2 + 4x - 11x - 44 \
&= x^2 - 7x - 44.
\end{aligned}
$$

Then, multiply through by $$ a $$:

$$
f(x) = a\left(x^2 - 7x - 44\right) = ax^2 - 7ax - 44a.
$$

Thus, the quadratic function in standard form is

$$
f(x)=ax^2-7ax-44a,
$$

with $$ a \neq 0 $$. 
If no vertical stretch or compression is specified (i.e., if $$ a=1 $$), then the equation becomes

$$
f(x)=x^2-7x-44.
$$
The quadratic function with solutions $$ x=11 $$ and $$ x=-4 $$ in standard form is: $$ f(x) = x^2 - 7x - 44 $$

Write the equation of a quadratic function with the given solutions. Use standard form \( f(x)=a x^{2}+b x+c \) \( x=11 \) and \( x=-4 \)