Solve x^2-115x=550

$$x ^ { 2 } - 115 x = 550$$

Solve for x

$$x = \frac{5 \sqrt{617} + 115}{2} \approx 119.598711742$$

$$x=\frac{115-5\sqrt{617}}{2}\approx -4.598711742$$

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.

$$x^{2}-115x=550$$

Subtract 550 from both sides of the equation.

$$x^{2}-115x-550=550-550$$

Subtracting 550 from itself leaves 0.

$$x^{2}-115x-550=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -115 for b, and -550 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-115\right)±\sqrt{\left(-115\right)^{2}-4\left(-550\right)}}{2}$$

Square -115.

$$x=\frac{-\left(-115\right)±\sqrt{13225-4\left(-550\right)}}{2}$$

Multiply -4 times -550.

$$x=\frac{-\left(-115\right)±\sqrt{13225+2200}}{2}$$

Add 13225 to 2200.

$$x=\frac{-\left(-115\right)±\sqrt{15425}}{2}$$

Take the square root of 15425.

$$x=\frac{-\left(-115\right)±5\sqrt{617}}{2}$$

The opposite of -115 is 115.

$$x=\frac{115±5\sqrt{617}}{2}$$

Now solve the equation x=\frac{115±5\sqrt{617}}{2} when ± is plus. Add 115 to 5\sqrt{617}.

$$x=\frac{5\sqrt{617}+115}{2}$$

Now solve the equation x=\frac{115±5\sqrt{617}}{2} when ± is minus. Subtract 5\sqrt{617} from 115.

$$x=\frac{115-5\sqrt{617}}{2}$$

The equation is now solved.

$$x=\frac{5\sqrt{617}+115}{2} x=\frac{115-5\sqrt{617}}{2}$$

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.

$$x^{2}-115x=550$$

Divide -115, the coefficient of the x term, by 2 to get -\frac{115}{2}. Then add the square of -\frac{115}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-115x+\left(-\frac{115}{2}\right)^{2}=550+\left(-\frac{115}{2}\right)^{2}$$

Square -\frac{115}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-115x+\frac{13225}{4}=550+\frac{13225}{4}$$

Add 550 to \frac{13225}{4}.

$$x^{2}-115x+\frac{13225}{4}=\frac{15425}{4}$$

Factor x^{2}-115x+\frac{13225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{115}{2}\right)^{2}=\frac{15425}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{115}{2}\right)^{2}}=\sqrt{\frac{15425}{4}}$$

Simplify.

$$x-\frac{115}{2}=\frac{5\sqrt{617}}{2} x-\frac{115}{2}=-\frac{5\sqrt{617}}{2}$$

Add \frac{115}{2} to both sides of the equation.

$$x=\frac{5\sqrt{617}+115}{2} x=\frac{115-5\sqrt{617}}{2}$$