Solve x^2-31x-276=0

$${ x }^{ 2 } -31x-276=0$$

Solve for x

$$x = \frac{\sqrt{2065} + 31}{2} \approx 38.221135535$$

$$x=\frac{31-\sqrt{2065}}{2}\approx -7.221135535$$

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.

$$x^{2}-31x-276=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -31 for b, and -276 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\left(-276\right)}}{2}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\left(-276\right)}}{2}$$

Multiply -4 times -276.

$$x=\frac{-\left(-31\right)±\sqrt{961+1104}}{2}$$

Add 961 to 1104.

$$x=\frac{-\left(-31\right)±\sqrt{2065}}{2}$$

The opposite of -31 is 31.

$$x=\frac{31±\sqrt{2065}}{2}$$

Now solve the equation x=\frac{31±\sqrt{2065}}{2} when ± is plus. Add 31 to \sqrt{2065}.

$$x=\frac{\sqrt{2065}+31}{2}$$

Now solve the equation x=\frac{31±\sqrt{2065}}{2} when ± is minus. Subtract \sqrt{2065} from 31.

$$x=\frac{31-\sqrt{2065}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2065}+31}{2} x=\frac{31-\sqrt{2065}}{2}$$

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.

$$x^{2}-31x-276=0$$

Add 276 to both sides of the equation.

$$x^{2}-31x-276-\left(-276\right)=-\left(-276\right)$$

Subtracting -276 from itself leaves 0.

$$x^{2}-31x=-\left(-276\right)$$

Subtract -276 from 0.

$$x^{2}-31x=276$$

Divide -31, the coefficient of the x term, by 2 to get -\frac{31}{2}. Then add the square of -\frac{31}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-31x+\left(-\frac{31}{2}\right)^{2}=276+\left(-\frac{31}{2}\right)^{2}$$

Square -\frac{31}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-31x+\frac{961}{4}=276+\frac{961}{4}$$

Add 276 to \frac{961}{4}.

$$x^{2}-31x+\frac{961}{4}=\frac{2065}{4}$$

Factor x^{2}-31x+\frac{961}{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}.

$$\left(x-\frac{31}{2}\right)^{2}=\frac{2065}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{2}\right)^{2}}=\sqrt{\frac{2065}{4}}$$

Simplify.

$$x-\frac{31}{2}=\frac{\sqrt{2065}}{2} x-\frac{31}{2}=-\frac{\sqrt{2065}}{2}$$

Add \frac{31}{2} to both sides of the equation.

$$x=\frac{\sqrt{2065}+31}{2} x=\frac{31-\sqrt{2065}}{2}$$