Solve x^2-90x-36

$${ x }^{ 2 } -90x-36$$

Factor

$$\left(x-\left(45-3\sqrt{229}\right)\right)\left(x-\left(3\sqrt{229}+45\right)\right)$$

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.

$$x^{2}-90x-36=0$$

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.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\left(-36\right)}}{2}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$x=\frac{-\left(-90\right)±\sqrt{8100+144}}{2}$$

Add 8100 to 144.

$$x=\frac{-\left(-90\right)±\sqrt{8244}}{2}$$

Take the square root of 8244.

$$x=\frac{-\left(-90\right)±6\sqrt{229}}{2}$$

The opposite of -90 is 90.

$$x=\frac{90±6\sqrt{229}}{2}$$

Now solve the equation x=\frac{90±6\sqrt{229}}{2} when ± is plus. Add 90 to 6\sqrt{229}.

$$x=\frac{6\sqrt{229}+90}{2}$$

Divide 90+6\sqrt{229} by 2.

$$x=3\sqrt{229}+45$$

Now solve the equation x=\frac{90±6\sqrt{229}}{2} when ± is minus. Subtract 6\sqrt{229} from 90.

$$x=\frac{90-6\sqrt{229}}{2}$$

Divide 90-6\sqrt{229} by 2.

$$x=45-3\sqrt{229}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 45+3\sqrt{229} for x_{1} and 45-3\sqrt{229} for x_{2}.

$$x^{2}-90x-36=\left(x-\left(3\sqrt{229}+45\right)\right)\left(x-\left(45-3\sqrt{229}\right)\right)$$

Evaluate

$$x^{2}-90x-36$$