Solve x^2-65x+512=0

$${ x }^{ 2 } -65x+512=0$$

Solve for x

$$x = \frac{\sqrt{2177} + 65}{2} \approx 55.829166295$$

$$x = \frac{65 - \sqrt{2177}}{2} \approx 9.170833705$$

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.

$$x^{2}-65x+512=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -65 for b, and 512 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-65\right)±\sqrt{\left(-65\right)^{2}-4\times 512}}{2}$$

Square -65.

$$x=\frac{-\left(-65\right)±\sqrt{4225-4\times 512}}{2}$$

Multiply -4 times 512.

$$x=\frac{-\left(-65\right)±\sqrt{4225-2048}}{2}$$

Add 4225 to -2048.

$$x=\frac{-\left(-65\right)±\sqrt{2177}}{2}$$

The opposite of -65 is 65.

$$x=\frac{65±\sqrt{2177}}{2}$$

Now solve the equation x=\frac{65±\sqrt{2177}}{2} when ± is plus. Add 65 to \sqrt{2177}.

$$x=\frac{\sqrt{2177}+65}{2}$$

Now solve the equation x=\frac{65±\sqrt{2177}}{2} when ± is minus. Subtract \sqrt{2177} from 65.

$$x=\frac{65-\sqrt{2177}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2177}+65}{2} x=\frac{65-\sqrt{2177}}{2}$$

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.

$$x^{2}-65x+512=0$$

Subtract 512 from both sides of the equation.

$$x^{2}-65x+512-512=-512$$

Subtracting 512 from itself leaves 0.

$$x^{2}-65x=-512$$

Divide -65, the coefficient of the x term, by 2 to get -\frac{65}{2}. Then add the square of -\frac{65}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-65x+\left(-\frac{65}{2}\right)^{2}=-512+\left(-\frac{65}{2}\right)^{2}$$

Square -\frac{65}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-65x+\frac{4225}{4}=-512+\frac{4225}{4}$$

Add -512 to \frac{4225}{4}.

$$x^{2}-65x+\frac{4225}{4}=\frac{2177}{4}$$

Factor x^{2}-65x+\frac{4225}{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}.

$$\left(x-\frac{65}{2}\right)^{2}=\frac{2177}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{65}{2}\right)^{2}}=\sqrt{\frac{2177}{4}}$$

Simplify.

$$x-\frac{65}{2}=\frac{\sqrt{2177}}{2} x-\frac{65}{2}=-\frac{\sqrt{2177}}{2}$$

Add \frac{65}{2} to both sides of the equation.

$$x=\frac{\sqrt{2177}+65}{2} x=\frac{65-\sqrt{2177}}{2}$$