Solve x^2-16x+61

$${ x }^{ 2 } -16x+61$$

Factor

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

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.

$$x^{2}-16x+61=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 61}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 61}}{2}$$

Multiply -4 times 61.

$$x=\frac{-\left(-16\right)±\sqrt{256-244}}{2}$$

Add 256 to -244.

$$x=\frac{-\left(-16\right)±\sqrt{12}}{2}$$

Take the square root of 12.

$$x=\frac{-\left(-16\right)±2\sqrt{3}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{3}}{2}$$

Now solve the equation x=\frac{16±2\sqrt{3}}{2} when ± is plus. Add 16 to 2\sqrt{3}.

$$x=\frac{2\sqrt{3}+16}{2}$$

Divide 16+2\sqrt{3} by 2.

$$x=\sqrt{3}+8$$

Now solve the equation x=\frac{16±2\sqrt{3}}{2} when ± is minus. Subtract 2\sqrt{3} from 16.

$$x=\frac{16-2\sqrt{3}}{2}$$

Divide 16-2\sqrt{3} by 2.

$$x=8-\sqrt{3}$$

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

$$x^{2}-16x+61=\left(x-\left(\sqrt{3}+8\right)\right)\left(x-\left(8-\sqrt{3}\right)\right)$$

Evaluate

$$x^{2}-16x+61$$