Solve x^2=(x-10)+24^2

$$x ^ { 2 } = ( x - 10 ) + 24 ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{2265} + 1}{2} \approx 24.296008069$$

$$x=\frac{1-\sqrt{2265}}{2}\approx -23.296008069$$

Steps Using the Quadratic Formula

Calculate 24 to the power of 2 and get 576.

$$x^{2}=x-10+576$$

Add -10 and 576 to get 566.

$$x^{2}=x+566$$

Subtract x from both sides.

$$x^{2}-x=566$$

Subtract 566 from both sides.

$$x^{2}-x-566=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -566 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-566\right)}}{2}$$

Multiply -4 times -566.

$$x=\frac{-\left(-1\right)±\sqrt{1+2264}}{2}$$

Add 1 to 2264.

$$x=\frac{-\left(-1\right)±\sqrt{2265}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{2265}}{2}$$

Now solve the equation x=\frac{1±\sqrt{2265}}{2} when ± is plus. Add 1 to \sqrt{2265}.

$$x=\frac{\sqrt{2265}+1}{2}$$

Now solve the equation x=\frac{1±\sqrt{2265}}{2} when ± is minus. Subtract \sqrt{2265} from 1.

$$x=\frac{1-\sqrt{2265}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2265}+1}{2} x=\frac{1-\sqrt{2265}}{2}$$

Steps for Completing the Square

Calculate 24 to the power of 2 and get 576.

$$x^{2}=x-10+576$$

Add -10 and 576 to get 566.

$$x^{2}=x+566$$

Subtract x from both sides.

$$x^{2}-x=566$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=566+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=566+\frac{1}{4}$$

Add 566 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{2265}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{2265}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{2265}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{2265}}{2} x-\frac{1}{2}=-\frac{\sqrt{2265}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{2265}+1}{2} x=\frac{1-\sqrt{2265}}{2}$$