Solve (76-4x)(8+2x)=1080

$$( 76 - 4 x ) ( 8 + 2 x ) = 1080$$

Solve for x (complex solution)

$$x=\frac{15+\sqrt{11}i}{2}\approx 7.5+1.658312395i$$

$$x=\frac{-\sqrt{11}i+15}{2}\approx 7.5-1.658312395i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 76-4x by 8+2x and combine like terms.

$$608+120x-8x^{2}=1080$$

Subtract 1080 from both sides.

$$608+120x-8x^{2}-1080=0$$

Subtract 1080 from 608 to get -472.

$$-472+120x-8x^{2}=0$$

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.

$$-8x^{2}+120x-472=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 120 for b, and -472 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-8\right)\left(-472\right)}}{2\left(-8\right)}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\left(-8\right)\left(-472\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-120±\sqrt{14400+32\left(-472\right)}}{2\left(-8\right)}$$

Multiply 32 times -472.

$$x=\frac{-120±\sqrt{14400-15104}}{2\left(-8\right)}$$

Add 14400 to -15104.

$$x=\frac{-120±\sqrt{-704}}{2\left(-8\right)}$$

Take the square root of -704.

$$x=\frac{-120±8\sqrt{11}i}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-120±8\sqrt{11}i}{-16}$$

Now solve the equation x=\frac{-120±8\sqrt{11}i}{-16} when ± is plus. Add -120 to 8i\sqrt{11}.

$$x=\frac{-120+8\sqrt{11}i}{-16}$$

Divide -120+8i\sqrt{11} by -16.

$$x=\frac{-\sqrt{11}i+15}{2}$$

Now solve the equation x=\frac{-120±8\sqrt{11}i}{-16} when ± is minus. Subtract 8i\sqrt{11} from -120.

$$x=\frac{-8\sqrt{11}i-120}{-16}$$

Divide -120-8i\sqrt{11} by -16.

$$x=\frac{15+\sqrt{11}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{11}i+15}{2} x=\frac{15+\sqrt{11}i}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 76-4x by 8+2x and combine like terms.

$$608+120x-8x^{2}=1080$$

Subtract 608 from both sides.

$$120x-8x^{2}=1080-608$$

Subtract 608 from 1080 to get 472.

$$120x-8x^{2}=472$$

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.

$$-8x^{2}+120x=472$$

Divide both sides by -8.

$$\frac{-8x^{2}+120x}{-8}=\frac{472}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{120}{-8}x=\frac{472}{-8}$$

Divide 120 by -8.

$$x^{2}-15x=\frac{472}{-8}$$

Divide 472 by -8.

$$x^{2}-15x=-59$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-59+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-59+\frac{225}{4}$$

Add -59 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=-\frac{11}{4}$$

Factor x^{2}-15x+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{2}\right)^{2}=-\frac{11}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{-\frac{11}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{\sqrt{11}i}{2} x-\frac{15}{2}=-\frac{\sqrt{11}i}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=\frac{15+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+15}{2}$$