Solve x(8-x)-24=0

$$x(8-x)-24=0$$

Solve for x (complex solution)

$$x=-2\sqrt{2}i+4\approx 4-2.828427125i$$

$$x=4+2\sqrt{2}i\approx 4+2.828427125i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 8-x.

$$8x-x^{2}-24=0$$

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.

$$-x^{2}+8x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 8 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\left(-24\right)}}{2\left(-1\right)}$$

Multiply 4 times -24.

$$x=\frac{-8±\sqrt{64-96}}{2\left(-1\right)}$$

Add 64 to -96.

$$x=\frac{-8±\sqrt{-32}}{2\left(-1\right)}$$

Take the square root of -32.

$$x=\frac{-8±4\sqrt{2}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±4\sqrt{2}i}{-2}$$

Now solve the equation x=\frac{-8±4\sqrt{2}i}{-2} when ± is plus. Add -8 to 4i\sqrt{2}.

$$x=\frac{-8+2^{\frac{5}{2}}i}{-2}$$

Divide -8+i\times 2^{\frac{5}{2}} by -2.

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

Now solve the equation x=\frac{-8±4\sqrt{2}i}{-2} when ± is minus. Subtract 4i\sqrt{2} from -8.

$$x=\frac{-2^{\frac{5}{2}}i-8}{-2}$$

Divide -8-i\times 2^{\frac{5}{2}} by -2.

$$x=4+2\sqrt{2}i$$

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply x by 8-x.

$$8x-x^{2}-24=0$$

Add 24 to both sides. Anything plus zero gives itself.

$$8x-x^{2}=24$$

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.

$$-x^{2}+8x=24$$

Divide both sides by -1.

$$\frac{-x^{2}+8x}{-1}=\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{8}{-1}x=\frac{24}{-1}$$

Divide 8 by -1.

$$x^{2}-8x=\frac{24}{-1}$$

Divide 24 by -1.

$$x^{2}-8x=-24$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-24+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-24+16$$

Add -24 to 16.

$$x^{2}-8x+16=-8$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=-8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-8}$$

Simplify.

$$x-4=2\sqrt{2}i x-4=-2\sqrt{2}i$$

Add 4 to both sides of the equation.

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