Solve x^2-24x=496

$$x ^ { 2 } - 24 x = 496$$

Solve for x

$$x=8\sqrt{10}+12\approx 37.298221281$$

$$x=12-8\sqrt{10}\approx -13.298221281$$

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.

$$x^{2}-24x=496$$

Subtract 496 from both sides of the equation.

$$x^{2}-24x-496=496-496$$

Subtracting 496 from itself leaves 0.

$$x^{2}-24x-496=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and -496 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-496\right)}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-496\right)}}{2}$$

Multiply -4 times -496.

$$x=\frac{-\left(-24\right)±\sqrt{576+1984}}{2}$$

Add 576 to 1984.

$$x=\frac{-\left(-24\right)±\sqrt{2560}}{2}$$

Take the square root of 2560.

$$x=\frac{-\left(-24\right)±16\sqrt{10}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±16\sqrt{10}}{2}$$

Now solve the equation x=\frac{24±16\sqrt{10}}{2} when ± is plus. Add 24 to 16\sqrt{10}.

$$x=\frac{16\sqrt{10}+24}{2}$$

Divide 24+16\sqrt{10} by 2.

$$x=8\sqrt{10}+12$$

Now solve the equation x=\frac{24±16\sqrt{10}}{2} when ± is minus. Subtract 16\sqrt{10} from 24.

$$x=\frac{24-16\sqrt{10}}{2}$$

Divide 24-16\sqrt{10} by 2.

$$x=12-8\sqrt{10}$$

The equation is now solved.

$$x=8\sqrt{10}+12 x=12-8\sqrt{10}$$

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.

$$x^{2}-24x=496$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=496+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=496+144$$

Add 496 to 144.

$$x^{2}-24x+144=640$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=640$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{640}$$

Simplify.

$$x-12=8\sqrt{10} x-12=-8\sqrt{10}$$

Add 12 to both sides of the equation.

$$x=8\sqrt{10}+12 x=12-8\sqrt{10}$$