Solve (1)=60(x+3)(x-2

$$(1)=60(x+3)(x-2$$

Solve for x

$$x=\frac{\sqrt{1410}}{15}-\frac{1}{2}\approx 2.003331114$$

$$x=-\frac{\sqrt{1410}}{15}-\frac{1}{2}\approx -3.003331114$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 60 by x+3.

$$1=\left(60x+180\right)\left(x-2\right)$$

Use the distributive property to multiply 60x+180 by x-2 and combine like terms.

$$1=60x^{2}+60x-360$$

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

$$60x^{2}+60x-360=1$$

Subtract 1 from both sides.

$$60x^{2}+60x-360-1=0$$

Subtract 1 from -360 to get -361.

$$60x^{2}+60x-361=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 60 for a, 60 for b, and -361 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 60\left(-361\right)}}{2\times 60}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\times 60\left(-361\right)}}{2\times 60}$$

Multiply -4 times 60.

$$x=\frac{-60±\sqrt{3600-240\left(-361\right)}}{2\times 60}$$

Multiply -240 times -361.

$$x=\frac{-60±\sqrt{3600+86640}}{2\times 60}$$

Add 3600 to 86640.

$$x=\frac{-60±\sqrt{90240}}{2\times 60}$$

Take the square root of 90240.

$$x=\frac{-60±8\sqrt{1410}}{2\times 60}$$

Multiply 2 times 60.

$$x=\frac{-60±8\sqrt{1410}}{120}$$

Now solve the equation x=\frac{-60±8\sqrt{1410}}{120} when ± is plus. Add -60 to 8\sqrt{1410}.

$$x=\frac{8\sqrt{1410}-60}{120}$$

Divide -60+8\sqrt{1410} by 120.

$$x=\frac{\sqrt{1410}}{15}-\frac{1}{2}$$

Now solve the equation x=\frac{-60±8\sqrt{1410}}{120} when ± is minus. Subtract 8\sqrt{1410} from -60.

$$x=\frac{-8\sqrt{1410}-60}{120}$$

Divide -60-8\sqrt{1410} by 120.

$$x=-\frac{\sqrt{1410}}{15}-\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1410}}{15}-\frac{1}{2} x=-\frac{\sqrt{1410}}{15}-\frac{1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 60 by x+3.

$$1=\left(60x+180\right)\left(x-2\right)$$

Use the distributive property to multiply 60x+180 by x-2 and combine like terms.

$$1=60x^{2}+60x-360$$

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

$$60x^{2}+60x-360=1$$

Add 360 to both sides.

$$60x^{2}+60x=1+360$$

Add 1 and 360 to get 361.

$$60x^{2}+60x=361$$

Divide both sides by 60.

$$\frac{60x^{2}+60x}{60}=\frac{361}{60}$$

Dividing by 60 undoes the multiplication by 60.

$$x^{2}+\frac{60}{60}x=\frac{361}{60}$$

Divide 60 by 60.

$$x^{2}+x=\frac{361}{60}$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{361}{60}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{361}{60}+\frac{1}{4}$$

Add \frac{361}{60} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{94}{15}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{94}{15}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{94}{15}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{1410}}{15} x+\frac{1}{2}=-\frac{\sqrt{1410}}{15}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{1410}}{15}-\frac{1}{2} x=-\frac{\sqrt{1410}}{15}-\frac{1}{2}$$