Solve (a^2+1)-16a=

$$( a ^ { 2 } + 1 ) - 16 a =$$

Factor

$$\left(a-\left(8-3\sqrt{7}\right)\right)\left(a-\left(3\sqrt{7}+8\right)\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$a^{2}-16a+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$a=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4}}{2}$$

Square -16.

$$a=\frac{-\left(-16\right)±\sqrt{256-4}}{2}$$

Add 256 to -4.

$$a=\frac{-\left(-16\right)±\sqrt{252}}{2}$$

Take the square root of 252.

$$a=\frac{-\left(-16\right)±6\sqrt{7}}{2}$$

The opposite of -16 is 16.

$$a=\frac{16±6\sqrt{7}}{2}$$

Now solve the equation a=\frac{16±6\sqrt{7}}{2} when ± is plus. Add 16 to 6\sqrt{7}.

$$a=\frac{6\sqrt{7}+16}{2}$$

Divide 16+6\sqrt{7} by 2.

$$a=3\sqrt{7}+8$$

Now solve the equation a=\frac{16±6\sqrt{7}}{2} when ± is minus. Subtract 6\sqrt{7} from 16.

$$a=\frac{16-6\sqrt{7}}{2}$$

Divide 16-6\sqrt{7} by 2.

$$a=8-3\sqrt{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8+3\sqrt{7} for x_{1} and 8-3\sqrt{7} for x_{2}.

$$a^{2}-16a+1=\left(a-\left(3\sqrt{7}+8\right)\right)\left(a-\left(8-3\sqrt{7}\right)\right)$$

Evaluate

$$a^{2}-16a+1$$