Question

Factor f(a) = 7a²– 16a + 4

Answer 1

Factor this expression means to find two numbers x and y such that we can write f(a) = (a+x)(a+y).

To do this, we use the quadratic formula

$\begin{gathered} a=\frac{-(-16)\pm\sqrt[]{(-16)^2-4(7)(4)}}{2(7)} \\ a=\frac{16\pm\sqrt[]{256-112}}{14} \\ a=\frac{16\pm\sqrt[]{144}}{14} \\ a=\frac{16\pm12}{14} \end{gathered}$

so, one answer is

$a=\frac{16+12}{14}=\frac{28}{14}=2$

and the other one

$a=\frac{16-12}{14}=\frac{4}{14}=\frac{2}{7}$

To find the factors we use the solutions

$\begin{gathered} a=2\text{ then a-2=0} \\ a-2\text{ is a factor} \end{gathered}$$\begin{gathered} a=\frac{2}{7}\text{ then 7a-2=0} \\ 7a-2\text{ is a factor} \end{gathered}$

Then, the factorization is

$7a^2-16a+4=(7a-2)(a-2)$