Solve (10+2x)(76-4x)=1080

$$(10+2x)(76-4x)=1080$$

Solve for x

$$x=4$$

$$x=10$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10+2x by 76-4x and combine like terms.

$$760+112x-8x^{2}=1080$$

Subtract 1080 from both sides.

$$760+112x-8x^{2}-1080=0$$

Subtract 1080 from 760 to get -320.

$$-320+112x-8x^{2}=0$$

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.

$$-8x^{2}+112x-320=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 112 for b, and -320 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112±\sqrt{112^{2}-4\left(-8\right)\left(-320\right)}}{2\left(-8\right)}$$

Square 112.

$$x=\frac{-112±\sqrt{12544-4\left(-8\right)\left(-320\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-112±\sqrt{12544+32\left(-320\right)}}{2\left(-8\right)}$$

Multiply 32 times -320.

$$x=\frac{-112±\sqrt{12544-10240}}{2\left(-8\right)}$$

Add 12544 to -10240.

$$x=\frac{-112±\sqrt{2304}}{2\left(-8\right)}$$

Take the square root of 2304.

$$x=\frac{-112±48}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-112±48}{-16}$$

Now solve the equation x=\frac{-112±48}{-16} when ± is plus. Add -112 to 48.

$$x=-\frac{64}{-16}$$

Divide -64 by -16.

$$x=4$$

Now solve the equation x=\frac{-112±48}{-16} when ± is minus. Subtract 48 from -112.

$$x=-\frac{160}{-16}$$

Divide -160 by -16.

$$x=10$$

The equation is now solved.

$$x=4 x=10$$

Steps for Completing the Square

Use the distributive property to multiply 10+2x by 76-4x and combine like terms.

$$760+112x-8x^{2}=1080$$

Subtract 760 from both sides.

$$112x-8x^{2}=1080-760$$

Subtract 760 from 1080 to get 320.

$$112x-8x^{2}=320$$

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.

$$-8x^{2}+112x=320$$

Divide both sides by -8.

$$\frac{-8x^{2}+112x}{-8}=\frac{320}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{112}{-8}x=\frac{320}{-8}$$

Divide 112 by -8.

$$x^{2}-14x=\frac{320}{-8}$$

Divide 320 by -8.

$$x^{2}-14x=-40$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-40+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-40+49$$

Add -40 to 49.

$$x^{2}-14x+49=9$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-7=3 x-7=-3$$

Add 7 to both sides of the equation.

$$x=10 x=4$$