Solve (95-x)(6+2x)=1120

$$( 95 - x ) ( 6 + 2 x ) = 1120$$

Solve for x

$$x=\sqrt{1841}+46\approx 88.906875906$$

$$x=46-\sqrt{1841}\approx 3.093124094$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 95-x by 6+2x and combine like terms.

$$570+184x-2x^{2}=1120$$

Subtract 1120 from both sides.

$$570+184x-2x^{2}-1120=0$$

Subtract 1120 from 570 to get -550.

$$-550+184x-2x^{2}=0$$

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.

$$-2x^{2}+184x-550=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 184 for b, and -550 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-184±\sqrt{184^{2}-4\left(-2\right)\left(-550\right)}}{2\left(-2\right)}$$

Square 184.

$$x=\frac{-184±\sqrt{33856-4\left(-2\right)\left(-550\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-184±\sqrt{33856+8\left(-550\right)}}{2\left(-2\right)}$$

Multiply 8 times -550.

$$x=\frac{-184±\sqrt{33856-4400}}{2\left(-2\right)}$$

Add 33856 to -4400.

$$x=\frac{-184±\sqrt{29456}}{2\left(-2\right)}$$

Take the square root of 29456.

$$x=\frac{-184±4\sqrt{1841}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-184±4\sqrt{1841}}{-4}$$

Now solve the equation x=\frac{-184±4\sqrt{1841}}{-4} when ± is plus. Add -184 to 4\sqrt{1841}.

$$x=\frac{4\sqrt{1841}-184}{-4}$$

Divide -184+4\sqrt{1841} by -4.

$$x=46-\sqrt{1841}$$

Now solve the equation x=\frac{-184±4\sqrt{1841}}{-4} when ± is minus. Subtract 4\sqrt{1841} from -184.

$$x=\frac{-4\sqrt{1841}-184}{-4}$$

Divide -184-4\sqrt{1841} by -4.

$$x=\sqrt{1841}+46$$

The equation is now solved.

$$x=46-\sqrt{1841} x=\sqrt{1841}+46$$

Steps for Completing the Square

Use the distributive property to multiply 95-x by 6+2x and combine like terms.

$$570+184x-2x^{2}=1120$$

Subtract 570 from both sides.

$$184x-2x^{2}=1120-570$$

Subtract 570 from 1120 to get 550.

$$184x-2x^{2}=550$$

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.

$$-2x^{2}+184x=550$$

Divide both sides by -2.

$$\frac{-2x^{2}+184x}{-2}=\frac{550}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{184}{-2}x=\frac{550}{-2}$$

Divide 184 by -2.

$$x^{2}-92x=\frac{550}{-2}$$

Divide 550 by -2.

$$x^{2}-92x=-275$$

Divide -92, the coefficient of the x term, by 2 to get -46. Then add the square of -46 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-92x+\left(-46\right)^{2}=-275+\left(-46\right)^{2}$$

Square -46.

$$x^{2}-92x+2116=-275+2116$$

Add -275 to 2116.

$$x^{2}-92x+2116=1841$$

Factor x^{2}-92x+2116. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-46\right)^{2}=1841$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-46\right)^{2}}=\sqrt{1841}$$

Simplify.

$$x-46=\sqrt{1841} x-46=-\sqrt{1841}$$

Add 46 to both sides of the equation.

$$x=\sqrt{1841}+46 x=46-\sqrt{1841}$$