Solve left(x+30right)left(2x+200right)-450=180

$$\left( x+30 \right) \left( 2x+200 \right) -450 = 180$$

Solve for x

$$x=2\sqrt{385}-65\approx -25.757166259$$

$$x=-2\sqrt{385}-65\approx -104.242833741$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+30 by 2x+200 and combine like terms.

$$2x^{2}+260x+6000-450=180$$

Subtract 450 from 6000 to get 5550.

$$2x^{2}+260x+5550=180$$

Subtract 180 from both sides.

$$2x^{2}+260x+5550-180=0$$

Subtract 180 from 5550 to get 5370.

$$2x^{2}+260x+5370=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 260 for b, and 5370 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-260±\sqrt{260^{2}-4\times 2\times 5370}}{2\times 2}$$

Square 260.

$$x=\frac{-260±\sqrt{67600-4\times 2\times 5370}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-260±\sqrt{67600-8\times 5370}}{2\times 2}$$

Multiply -8 times 5370.

$$x=\frac{-260±\sqrt{67600-42960}}{2\times 2}$$

Add 67600 to -42960.

$$x=\frac{-260±\sqrt{24640}}{2\times 2}$$

Take the square root of 24640.

$$x=\frac{-260±8\sqrt{385}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-260±8\sqrt{385}}{4}$$

Now solve the equation x=\frac{-260±8\sqrt{385}}{4} when ± is plus. Add -260 to 8\sqrt{385}.

$$x=\frac{8\sqrt{385}-260}{4}$$

Divide -260+8\sqrt{385} by 4.

$$x=2\sqrt{385}-65$$

Now solve the equation x=\frac{-260±8\sqrt{385}}{4} when ± is minus. Subtract 8\sqrt{385} from -260.

$$x=\frac{-8\sqrt{385}-260}{4}$$

Divide -260-8\sqrt{385} by 4.

$$x=-2\sqrt{385}-65$$

The equation is now solved.

$$x=2\sqrt{385}-65 x=-2\sqrt{385}-65$$

Steps for Completing the Square

Use the distributive property to multiply x+30 by 2x+200 and combine like terms.

$$2x^{2}+260x+6000-450=180$$

Subtract 450 from 6000 to get 5550.

$$2x^{2}+260x+5550=180$$

Subtract 5550 from both sides.

$$2x^{2}+260x=180-5550$$

Subtract 5550 from 180 to get -5370.

$$2x^{2}+260x=-5370$$

Divide both sides by 2.

$$\frac{2x^{2}+260x}{2}=-\frac{5370}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{260}{2}x=-\frac{5370}{2}$$

Divide 260 by 2.

$$x^{2}+130x=-\frac{5370}{2}$$

Divide -5370 by 2.

$$x^{2}+130x=-2685$$

Divide 130, the coefficient of the x term, by 2 to get 65. Then add the square of 65 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+130x+65^{2}=-2685+65^{2}$$

Square 65.

$$x^{2}+130x+4225=-2685+4225$$

Add -2685 to 4225.

$$x^{2}+130x+4225=1540$$

Factor x^{2}+130x+4225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+65\right)^{2}=1540$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+65\right)^{2}}=\sqrt{1540}$$

Simplify.

$$x+65=2\sqrt{385} x+65=-2\sqrt{385}$$

Subtract 65 from both sides of the equation.

$$x=2\sqrt{385}-65 x=-2\sqrt{385}-65$$