Solve x^2-2x+7/16

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

Factor

$$\frac{\left(4x-7\right)\left(4x-1\right)}{16}$$

Solution Steps

Factor out \frac{1}{16}.

$$\frac{16x^{2}-32x+7}{16}$$

Consider 16x^{2}-32x+7. Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=-32 ab=16\times 7=112$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 112.

$$-1,-112 -2,-56 -4,-28 -7,-16 -8,-14$$

Calculate the sum for each pair.

$$-1-112=-113 -2-56=-58 -4-28=-32 -7-16=-23 -8-14=-22$$

The solution is the pair that gives sum -32.

$$a=-28 b=-4$$

Rewrite 16x^{2}-32x+7 as \left(16x^{2}-28x\right)+\left(-4x+7\right).

$$\left(16x^{2}-28x\right)+\left(-4x+7\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(4x-7\right)-\left(4x-7\right)$$

Factor out common term 4x-7 by using distributive property.

$$\left(4x-7\right)\left(4x-1\right)$$

Rewrite the complete factored expression.

$$\frac{\left(4x-7\right)\left(4x-1\right)}{16}$$

Evaluate

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