Question

Factorize:
$x^2-2x+\frac{7}{16}$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ {x}^2-2x+\frac{7}{16} \\ =\frac{16x^2-32x+7}{16} \\ =\frac{1}{16}\left(16x^2-32x+7\right) \end{gathered}$$

Now, we have to split ($-$32) into two numbers such that their sum is ($-$32) and their product is 112, i.e., $16\times 7$.
Clearly, $\left(-4\right)+\left(-28\right)=-32\mathrm{and}\left(-4\right)\times \left(-28\right)=112$.

$$\begin{gathered} \therefore x^2-2x+\frac{7}{16}=\frac{1}{16}(16x^2-32x+7) \\ =\frac{1}{16}(16x^2-4x-28x+7) \\ =\frac{1}{16}\left[4x(4x-1)-7(4x-1)\right] \\ =\frac{1}{16}(4x-1)(4x-7) \end{gathered}$$