Solve (x-2)*180*x=144

$$(x-2) \times 180 \times x=144$$

Solve for x

$$x=\frac{3\sqrt{5}}{5}+1\approx 2.341640786$$

$$x=-\frac{3\sqrt{5}}{5}+1\approx -0.341640786$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-2 by 180.

$$\left(180x-360\right)x=144$$

Use the distributive property to multiply 180x-360 by x.

$$180x^{2}-360x=144$$

Subtract 144 from both sides.

$$180x^{2}-360x-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 180 for a, -360 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-360\right)±\sqrt{\left(-360\right)^{2}-4\times 180\left(-144\right)}}{2\times 180}$$

Square -360.

$$x=\frac{-\left(-360\right)±\sqrt{129600-4\times 180\left(-144\right)}}{2\times 180}$$

Multiply -4 times 180.

$$x=\frac{-\left(-360\right)±\sqrt{129600-720\left(-144\right)}}{2\times 180}$$

Multiply -720 times -144.

$$x=\frac{-\left(-360\right)±\sqrt{129600+103680}}{2\times 180}$$

Add 129600 to 103680.

$$x=\frac{-\left(-360\right)±\sqrt{233280}}{2\times 180}$$

Take the square root of 233280.

$$x=\frac{-\left(-360\right)±216\sqrt{5}}{2\times 180}$$

The opposite of -360 is 360.

$$x=\frac{360±216\sqrt{5}}{2\times 180}$$

Multiply 2 times 180.

$$x=\frac{360±216\sqrt{5}}{360}$$

Now solve the equation x=\frac{360±216\sqrt{5}}{360} when ± is plus. Add 360 to 216\sqrt{5}.

$$x=\frac{216\sqrt{5}+360}{360}$$

Divide 360+216\sqrt{5} by 360.

$$x=\frac{3\sqrt{5}}{5}+1$$

Now solve the equation x=\frac{360±216\sqrt{5}}{360} when ± is minus. Subtract 216\sqrt{5} from 360.

$$x=\frac{360-216\sqrt{5}}{360}$$

Divide 360-216\sqrt{5} by 360.

$$x=-\frac{3\sqrt{5}}{5}+1$$

The equation is now solved.

$$x=\frac{3\sqrt{5}}{5}+1 x=-\frac{3\sqrt{5}}{5}+1$$

Steps for Completing the Square

Use the distributive property to multiply x-2 by 180.

$$\left(180x-360\right)x=144$$

Use the distributive property to multiply 180x-360 by x.

$$180x^{2}-360x=144$$

Divide both sides by 180.

$$\frac{180x^{2}-360x}{180}=\frac{144}{180}$$

Dividing by 180 undoes the multiplication by 180.

$$x^{2}+\left(-\frac{360}{180}\right)x=\frac{144}{180}$$

Divide -360 by 180.

$$x^{2}-2x=\frac{144}{180}$$

Reduce the fraction \frac{144}{180} to lowest terms by extracting and canceling out 36.

$$x^{2}-2x=\frac{4}{5}$$

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

$$x^{2}-2x+1=\frac{4}{5}+1$$

Add \frac{4}{5} to 1.

$$x^{2}-2x+1=\frac{9}{5}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{9}{5}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{9}{5}}$$

Simplify.

$$x-1=\frac{3\sqrt{5}}{5} x-1=-\frac{3\sqrt{5}}{5}$$

Add 1 to both sides of the equation.

$$x=\frac{3\sqrt{5}}{5}+1 x=-\frac{3\sqrt{5}}{5}+1$$