Solve 450pi=2pix(x+15)

$$450 \pi =2 \pi x(x+15)$$

Solve for x

$$x = \frac{15 \sqrt{5} - 15}{2} \approx 9.270509831$$

$$x=\frac{-15\sqrt{5}-15}{2}\approx -24.270509831$$

Steps Using the Quadratic Formula

Cancel out \pi on both sides.

$$450=2x\left(x+15\right)$$

Use the distributive property to multiply 2x by x+15.

$$450=2x^{2}+30x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+30x=450$$

Subtract 450 from both sides.

$$2x^{2}+30x-450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 30 for b, and -450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 2\left(-450\right)}}{2\times 2}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 2\left(-450\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-30±\sqrt{900-8\left(-450\right)}}{2\times 2}$$

Multiply -8 times -450.

$$x=\frac{-30±\sqrt{900+3600}}{2\times 2}$$

Add 900 to 3600.

$$x=\frac{-30±\sqrt{4500}}{2\times 2}$$

Take the square root of 4500.

$$x=\frac{-30±30\sqrt{5}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-30±30\sqrt{5}}{4}$$

Now solve the equation x=\frac{-30±30\sqrt{5}}{4} when ± is plus. Add -30 to 30\sqrt{5}.

$$x=\frac{30\sqrt{5}-30}{4}$$

Divide -30+30\sqrt{5} by 4.

$$x=\frac{15\sqrt{5}-15}{2}$$

Now solve the equation x=\frac{-30±30\sqrt{5}}{4} when ± is minus. Subtract 30\sqrt{5} from -30.

$$x=\frac{-30\sqrt{5}-30}{4}$$

Divide -30-30\sqrt{5} by 4.

$$x=\frac{-15\sqrt{5}-15}{2}$$

The equation is now solved.

$$x=\frac{15\sqrt{5}-15}{2} x=\frac{-15\sqrt{5}-15}{2}$$

Steps for Completing the Square

Cancel out \pi on both sides.

$$450=2x\left(x+15\right)$$

Use the distributive property to multiply 2x by x+15.

$$450=2x^{2}+30x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+30x=450$$

Divide both sides by 2.

$$\frac{2x^{2}+30x}{2}=\frac{450}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{30}{2}x=\frac{450}{2}$$

Divide 30 by 2.

$$x^{2}+15x=\frac{450}{2}$$

Divide 450 by 2.

$$x^{2}+15x=225$$

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.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=225+\left(\frac{15}{2}\right)^{2}$$

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

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

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

$$x^{2}+15x+\frac{225}{4}=\frac{1125}{4}$$

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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{1125}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1125}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{15\sqrt{5}}{2} x+\frac{15}{2}=-\frac{15\sqrt{5}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=\frac{15\sqrt{5}-15}{2} x=\frac{-15\sqrt{5}-15}{2}$$