Solve 300=-x^2+24x

$$300 = - x ^ { 2 } + 24 x$$

Solve for x (complex solution)

$$x=12+2\sqrt{39}i\approx 12+12.489995997i$$

$$x=-2\sqrt{39}i+12\approx 12-12.489995997i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-x^{2}+24x=300$$

Subtract 300 from both sides.

$$-x^{2}+24x-300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and -300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-300\right)}}{2\left(-1\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-1\right)\left(-300\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-24±\sqrt{576+4\left(-300\right)}}{2\left(-1\right)}$$

Multiply 4 times -300.

$$x=\frac{-24±\sqrt{576-1200}}{2\left(-1\right)}$$

Add 576 to -1200.

$$x=\frac{-24±\sqrt{-624}}{2\left(-1\right)}$$

Take the square root of -624.

$$x=\frac{-24±4\sqrt{39}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-24±4\sqrt{39}i}{-2}$$

Now solve the equation x=\frac{-24±4\sqrt{39}i}{-2} when ± is plus. Add -24 to 4i\sqrt{39}.

$$x=\frac{-24+4\sqrt{39}i}{-2}$$

Divide -24+4i\sqrt{39} by -2.

$$x=-2\sqrt{39}i+12$$

Now solve the equation x=\frac{-24±4\sqrt{39}i}{-2} when ± is minus. Subtract 4i\sqrt{39} from -24.

$$x=\frac{-4\sqrt{39}i-24}{-2}$$

Divide -24-4i\sqrt{39} by -2.

$$x=12+2\sqrt{39}i$$

The equation is now solved.

$$x=-2\sqrt{39}i+12 x=12+2\sqrt{39}i$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-x^{2}+24x=300$$

Divide both sides by -1.

$$\frac{-x^{2}+24x}{-1}=\frac{300}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{24}{-1}x=\frac{300}{-1}$$

Divide 24 by -1.

$$x^{2}-24x=\frac{300}{-1}$$

Divide 300 by -1.

$$x^{2}-24x=-300$$

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.

$$x^{2}-24x+\left(-12\right)^{2}=-300+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-300+144$$

Add -300 to 144.

$$x^{2}-24x+144=-156$$

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}.

$$\left(x-12\right)^{2}=-156$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{-156}$$

Simplify.

$$x-12=2\sqrt{39}i x-12=-2\sqrt{39}i$$

Add 12 to both sides of the equation.

$$x=12+2\sqrt{39}i x=-2\sqrt{39}i+12$$