Solve 24x-x^2=360

$$24 x - x ^ { 2 } = 360$$

Solve for x (complex solution)

$$x=-6\sqrt{6}i+12\approx 12-14.696938457i$$

$$x=12+6\sqrt{6}i\approx 12+14.696938457i$$

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.

$$-x^{2}+24x=360$$

Subtract 360 from both sides of the equation.

$$-x^{2}+24x-360=360-360$$

Subtracting 360 from itself leaves 0.

$$-x^{2}+24x-360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and -360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-360\right)}}{2\left(-1\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-1\right)\left(-360\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-24±\sqrt{576+4\left(-360\right)}}{2\left(-1\right)}$$

Multiply 4 times -360.

$$x=\frac{-24±\sqrt{576-1440}}{2\left(-1\right)}$$

Add 576 to -1440.

$$x=\frac{-24±\sqrt{-864}}{2\left(-1\right)}$$

Take the square root of -864.

$$x=\frac{-24±12\sqrt{6}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-24±12\sqrt{6}i}{-2}$$

Now solve the equation x=\frac{-24±12\sqrt{6}i}{-2} when ± is plus. Add -24 to 12i\sqrt{6}.

$$x=\frac{-24+12\sqrt{6}i}{-2}$$

Divide -24+12i\sqrt{6} by -2.

$$x=-6\sqrt{6}i+12$$

Now solve the equation x=\frac{-24±12\sqrt{6}i}{-2} when ± is minus. Subtract 12i\sqrt{6} from -24.

$$x=\frac{-12\sqrt{6}i-24}{-2}$$

Divide -24-12i\sqrt{6} by -2.

$$x=12+6\sqrt{6}i$$

The equation is now solved.

$$x=-6\sqrt{6}i+12 x=12+6\sqrt{6}i$$

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.

$$-x^{2}+24x=360$$

Divide both sides by -1.

$$\frac{-x^{2}+24x}{-1}=\frac{360}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{24}{-1}x=\frac{360}{-1}$$

Divide 24 by -1.

$$x^{2}-24x=\frac{360}{-1}$$

Divide 360 by -1.

$$x^{2}-24x=-360$$

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.

$$x^{2}-24x+\left(-12\right)^{2}=-360+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-360+144$$

Add -360 to 144.

$$x^{2}-24x+144=-216$$

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}.

$$\left(x-12\right)^{2}=-216$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{-216}$$

Simplify.

$$x-12=6\sqrt{6}i x-12=-6\sqrt{6}i$$

Add 12 to both sides of the equation.

$$x=12+6\sqrt{6}i x=-6\sqrt{6}i+12$$