Solve (80-4x)(2x+8)=18800

$$( 80 - 4 x ) ( 2 x + 8 ) = 18800$$

Solve for x (complex solution)

$$x=8+\sqrt{2206}i\approx 8+46.968074263i$$

$$x=-\sqrt{2206}i+8\approx 8-46.968074263i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 80-4x by 2x+8 and combine like terms.

$$128x+640-8x^{2}=18800$$

Subtract 18800 from both sides.

$$128x+640-8x^{2}-18800=0$$

Subtract 18800 from 640 to get -18160.

$$128x-18160-8x^{2}=0$$

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.

$$-8x^{2}+128x-18160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 128 for b, and -18160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-128±\sqrt{128^{2}-4\left(-8\right)\left(-18160\right)}}{2\left(-8\right)}$$

Square 128.

$$x=\frac{-128±\sqrt{16384-4\left(-8\right)\left(-18160\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-128±\sqrt{16384+32\left(-18160\right)}}{2\left(-8\right)}$$

Multiply 32 times -18160.

$$x=\frac{-128±\sqrt{16384-581120}}{2\left(-8\right)}$$

Add 16384 to -581120.

$$x=\frac{-128±\sqrt{-564736}}{2\left(-8\right)}$$

Take the square root of -564736.

$$x=\frac{-128±16\sqrt{2206}i}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-128±16\sqrt{2206}i}{-16}$$

Now solve the equation x=\frac{-128±16\sqrt{2206}i}{-16} when ± is plus. Add -128 to 16i\sqrt{2206}.

$$x=\frac{-128+16\sqrt{2206}i}{-16}$$

Divide -128+16i\sqrt{2206} by -16.

$$x=-\sqrt{2206}i+8$$

Now solve the equation x=\frac{-128±16\sqrt{2206}i}{-16} when ± is minus. Subtract 16i\sqrt{2206} from -128.

$$x=\frac{-16\sqrt{2206}i-128}{-16}$$

Divide -128-16i\sqrt{2206} by -16.

$$x=8+\sqrt{2206}i$$

The equation is now solved.

$$x=-\sqrt{2206}i+8 x=8+\sqrt{2206}i$$

Steps for Completing the Square

Use the distributive property to multiply 80-4x by 2x+8 and combine like terms.

$$128x+640-8x^{2}=18800$$

Subtract 640 from both sides.

$$128x-8x^{2}=18800-640$$

Subtract 640 from 18800 to get 18160.

$$128x-8x^{2}=18160$$

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.

$$-8x^{2}+128x=18160$$

Divide both sides by -8.

$$\frac{-8x^{2}+128x}{-8}=\frac{18160}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{128}{-8}x=\frac{18160}{-8}$$

Divide 128 by -8.

$$x^{2}-16x=\frac{18160}{-8}$$

Divide 18160 by -8.

$$x^{2}-16x=-2270$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-2270+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-2270+64$$

Add -2270 to 64.

$$x^{2}-16x+64=-2206$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=-2206$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{-2206}$$

Simplify.

$$x-8=\sqrt{2206}i x-8=-\sqrt{2206}i$$

Add 8 to both sides of the equation.

$$x=8+\sqrt{2206}i x=-\sqrt{2206}i+8$$