Solve x^2-90x-800=0

$${ x }^{ 2 } -90x-800 = 0$$

Solve for x

$$x=5\sqrt{113}+45\approx 98.150729064$$

$$x=45-5\sqrt{113}\approx -8.150729064$$

Steps Using the Quadratic Formula

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.

$$x^{2}-90x-800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -90 for b, and -800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\left(-800\right)}}{2}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\left(-800\right)}}{2}$$

Multiply -4 times -800.

$$x=\frac{-\left(-90\right)±\sqrt{8100+3200}}{2}$$

Add 8100 to 3200.

$$x=\frac{-\left(-90\right)±\sqrt{11300}}{2}$$

Take the square root of 11300.

$$x=\frac{-\left(-90\right)±10\sqrt{113}}{2}$$

The opposite of -90 is 90.

$$x=\frac{90±10\sqrt{113}}{2}$$

Now solve the equation x=\frac{90±10\sqrt{113}}{2} when ± is plus. Add 90 to 10\sqrt{113}.

$$x=\frac{10\sqrt{113}+90}{2}$$

Divide 90+10\sqrt{113} by 2.

$$x=5\sqrt{113}+45$$

Now solve the equation x=\frac{90±10\sqrt{113}}{2} when ± is minus. Subtract 10\sqrt{113} from 90.

$$x=\frac{90-10\sqrt{113}}{2}$$

Divide 90-10\sqrt{113} by 2.

$$x=45-5\sqrt{113}$$

The equation is now solved.

$$x=5\sqrt{113}+45 x=45-5\sqrt{113}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-90x-800=0$$

Add 800 to both sides of the equation.

$$x^{2}-90x-800-\left(-800\right)=-\left(-800\right)$$

Subtracting -800 from itself leaves 0.

$$x^{2}-90x=-\left(-800\right)$$

Subtract -800 from 0.

$$x^{2}-90x=800$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=800+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=800+2025$$

Add 800 to 2025.

$$x^{2}-90x+2025=2825$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=2825$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{2825}$$

Simplify.

$$x-45=5\sqrt{113} x-45=-5\sqrt{113}$$

Add 45 to both sides of the equation.

$$x=5\sqrt{113}+45 x=45-5\sqrt{113}$$