Solve -x^2+16x-51

$$- { x }^{ 2 } +16x-51$$

Factor

$$-\left(x-\left(8-\sqrt{13}\right)\right)\left(x-\left(\sqrt{13}+8\right)\right)$$

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.

$$-x^{2}+16x-51=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-51\right)}}{2\left(-1\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\left(-51\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-16±\sqrt{256+4\left(-51\right)}}{2\left(-1\right)}$$

Multiply 4 times -51.

$$x=\frac{-16±\sqrt{256-204}}{2\left(-1\right)}$$

Add 256 to -204.

$$x=\frac{-16±\sqrt{52}}{2\left(-1\right)}$$

Take the square root of 52.

$$x=\frac{-16±2\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-16±2\sqrt{13}}{-2}$$

Now solve the equation x=\frac{-16±2\sqrt{13}}{-2} when ± is plus. Add -16 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}-16}{-2}$$

Divide -16+2\sqrt{13} by -2.

$$x=8-\sqrt{13}$$

Now solve the equation x=\frac{-16±2\sqrt{13}}{-2} when ± is minus. Subtract 2\sqrt{13} from -16.

$$x=\frac{-2\sqrt{13}-16}{-2}$$

Divide -16-2\sqrt{13} by -2.

$$x=\sqrt{13}+8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8-\sqrt{13} for x_{1} and 8+\sqrt{13} for x_{2}.

$$-x^{2}+16x-51=-\left(x-\left(8-\sqrt{13}\right)\right)\left(x-\left(\sqrt{13}+8\right)\right)$$

Evaluate

$$-x^{2}+16x-51$$