Solve 360/x+156=(x-2)*180

$$\frac{ 360 }{ x } +156=(x-2) \times 180$$

Solve for x

$$x = \frac{\sqrt{3649} + 43}{30} \approx 3.446898441$$

$$x=\frac{43-\sqrt{3649}}{30}\approx -0.580231775$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$360+x\times 156=\left(x-2\right)\times 180x$$

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

$$360+x\times 156=\left(180x-360\right)x$$

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

$$360+x\times 156=180x^{2}-360x$$

Subtract 180x^{2} from both sides.

$$360+x\times 156-180x^{2}=-360x$$

Add 360x to both sides.

$$360+x\times 156-180x^{2}+360x=0$$

Combine x\times 156 and 360x to get 516x.

$$360+516x-180x^{2}=0$$

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.

$$-180x^{2}+516x+360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -180 for a, 516 for b, and 360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-516±\sqrt{516^{2}-4\left(-180\right)\times 360}}{2\left(-180\right)}$$

Square 516.

$$x=\frac{-516±\sqrt{266256-4\left(-180\right)\times 360}}{2\left(-180\right)}$$

Multiply -4 times -180.

$$x=\frac{-516±\sqrt{266256+720\times 360}}{2\left(-180\right)}$$

Multiply 720 times 360.

$$x=\frac{-516±\sqrt{266256+259200}}{2\left(-180\right)}$$

Add 266256 to 259200.

$$x=\frac{-516±\sqrt{525456}}{2\left(-180\right)}$$

Take the square root of 525456.

$$x=\frac{-516±12\sqrt{3649}}{2\left(-180\right)}$$

Multiply 2 times -180.

$$x=\frac{-516±12\sqrt{3649}}{-360}$$

Now solve the equation x=\frac{-516±12\sqrt{3649}}{-360} when ± is plus. Add -516 to 12\sqrt{3649}.

$$x=\frac{12\sqrt{3649}-516}{-360}$$

Divide -516+12\sqrt{3649} by -360.

$$x=\frac{43-\sqrt{3649}}{30}$$

Now solve the equation x=\frac{-516±12\sqrt{3649}}{-360} when ± is minus. Subtract 12\sqrt{3649} from -516.

$$x=\frac{-12\sqrt{3649}-516}{-360}$$

Divide -516-12\sqrt{3649} by -360.

$$x=\frac{\sqrt{3649}+43}{30}$$

The equation is now solved.

$$x=\frac{43-\sqrt{3649}}{30} x=\frac{\sqrt{3649}+43}{30}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$360+x\times 156=\left(x-2\right)\times 180x$$

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

$$360+x\times 156=\left(180x-360\right)x$$

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

$$360+x\times 156=180x^{2}-360x$$

Subtract 180x^{2} from both sides.

$$360+x\times 156-180x^{2}=-360x$$

Add 360x to both sides.

$$360+x\times 156-180x^{2}+360x=0$$

Combine x\times 156 and 360x to get 516x.

$$360+516x-180x^{2}=0$$

Subtract 360 from both sides. Anything subtracted from zero gives its negation.

$$516x-180x^{2}=-360$$

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.

$$-180x^{2}+516x=-360$$

Divide both sides by -180.

$$\frac{-180x^{2}+516x}{-180}=-\frac{360}{-180}$$

Dividing by -180 undoes the multiplication by -180.

$$x^{2}+\frac{516}{-180}x=-\frac{360}{-180}$$

Reduce the fraction \frac{516}{-180} to lowest terms by extracting and canceling out 12.

$$x^{2}-\frac{43}{15}x=-\frac{360}{-180}$$

Divide -360 by -180.

$$x^{2}-\frac{43}{15}x=2$$

Divide -\frac{43}{15}, the coefficient of the x term, by 2 to get -\frac{43}{30}. Then add the square of -\frac{43}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{43}{15}x+\left(-\frac{43}{30}\right)^{2}=2+\left(-\frac{43}{30}\right)^{2}$$

Square -\frac{43}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{43}{15}x+\frac{1849}{900}=2+\frac{1849}{900}$$

Add 2 to \frac{1849}{900}.

$$x^{2}-\frac{43}{15}x+\frac{1849}{900}=\frac{3649}{900}$$

Factor x^{2}-\frac{43}{15}x+\frac{1849}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{43}{30}\right)^{2}=\frac{3649}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{43}{30}\right)^{2}}=\sqrt{\frac{3649}{900}}$$

Simplify.

$$x-\frac{43}{30}=\frac{\sqrt{3649}}{30} x-\frac{43}{30}=-\frac{\sqrt{3649}}{30}$$

Add \frac{43}{30} to both sides of the equation.

$$x=\frac{\sqrt{3649}+43}{30} x=\frac{43-\sqrt{3649}}{30}$$