Solve -x^2-6x+36

$$- { x }^{ 2 } -6x+36$$

Factor

$$-\left(x-\left(-3\sqrt{5}-3\right)\right)\left(x-\left(3\sqrt{5}-3\right)\right)$$

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.

$$-x^{2}-6x+36=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 36}}{2\left(-1\right)}$$

Multiply 4 times 36.

$$x=\frac{-\left(-6\right)±\sqrt{36+144}}{2\left(-1\right)}$$

Add 36 to 144.

$$x=\frac{-\left(-6\right)±\sqrt{180}}{2\left(-1\right)}$$

Take the square root of 180.

$$x=\frac{-\left(-6\right)±6\sqrt{5}}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±6\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±6\sqrt{5}}{-2}$$

Now solve the equation x=\frac{6±6\sqrt{5}}{-2} when ± is plus. Add 6 to 6\sqrt{5}.

$$x=\frac{6\sqrt{5}+6}{-2}$$

Divide 6+6\sqrt{5} by -2.

$$x=-3\sqrt{5}-3$$

Now solve the equation x=\frac{6±6\sqrt{5}}{-2} when ± is minus. Subtract 6\sqrt{5} from 6.

$$x=\frac{6-6\sqrt{5}}{-2}$$

Divide 6-6\sqrt{5} by -2.

$$x=3\sqrt{5}-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3-3\sqrt{5} for x_{1} and -3+3\sqrt{5} for x_{2}.

$$-x^{2}-6x+36=-\left(x-\left(-3\sqrt{5}-3\right)\right)\left(x-\left(3\sqrt{5}-3\right)\right)$$

Evaluate

$$36-6x-x^{2}$$