Solve 72/x-72/x+10=36

$$\frac { 72 } { x } - \frac { 72 } { x + 10 } = 36$$

Solve for x

$$x=3\sqrt{5}-5\approx 1.708203932$$

$$x=-3\sqrt{5}-5\approx -11.708203932$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x,x+10.

$$\left(x+10\right)\times 72-x\times 72=36x\left(x+10\right)$$

Use the distributive property to multiply x+10 by 72.

$$72x+720-x\times 72=36x\left(x+10\right)$$

Use the distributive property to multiply 36x by x+10.

$$72x+720-x\times 72=36x^{2}+360x$$

Subtract 36x^{2} from both sides.

$$72x+720-x\times 72-36x^{2}=360x$$

Subtract 360x from both sides.

$$72x+720-x\times 72-36x^{2}-360x=0$$

Combine 72x and -360x to get -288x.

$$-288x+720-x\times 72-36x^{2}=0$$

Multiply -1 and 72 to get -72.

$$-288x+720-72x-36x^{2}=0$$

Combine -288x and -72x to get -360x.

$$-360x+720-36x^{2}=0$$

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.

$$-36x^{2}-360x+720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -36 for a, -360 for b, and 720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-360\right)±\sqrt{\left(-360\right)^{2}-4\left(-36\right)\times 720}}{2\left(-36\right)}$$

Square -360.

$$x=\frac{-\left(-360\right)±\sqrt{129600-4\left(-36\right)\times 720}}{2\left(-36\right)}$$

Multiply -4 times -36.

$$x=\frac{-\left(-360\right)±\sqrt{129600+144\times 720}}{2\left(-36\right)}$$

Multiply 144 times 720.

$$x=\frac{-\left(-360\right)±\sqrt{129600+103680}}{2\left(-36\right)}$$

Add 129600 to 103680.

$$x=\frac{-\left(-360\right)±\sqrt{233280}}{2\left(-36\right)}$$

Take the square root of 233280.

$$x=\frac{-\left(-360\right)±216\sqrt{5}}{2\left(-36\right)}$$

The opposite of -360 is 360.

$$x=\frac{360±216\sqrt{5}}{2\left(-36\right)}$$

Multiply 2 times -36.

$$x=\frac{360±216\sqrt{5}}{-72}$$

Now solve the equation x=\frac{360±216\sqrt{5}}{-72} when ± is plus. Add 360 to 216\sqrt{5}.

$$x=\frac{216\sqrt{5}+360}{-72}$$

Divide 360+216\sqrt{5} by -72.

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

Now solve the equation x=\frac{360±216\sqrt{5}}{-72} when ± is minus. Subtract 216\sqrt{5} from 360.

$$x=\frac{360-216\sqrt{5}}{-72}$$

Divide 360-216\sqrt{5} by -72.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x,x+10.

$$\left(x+10\right)\times 72-x\times 72=36x\left(x+10\right)$$

Use the distributive property to multiply x+10 by 72.

$$72x+720-x\times 72=36x\left(x+10\right)$$

Use the distributive property to multiply 36x by x+10.

$$72x+720-x\times 72=36x^{2}+360x$$

Subtract 36x^{2} from both sides.

$$72x+720-x\times 72-36x^{2}=360x$$

Subtract 360x from both sides.

$$72x+720-x\times 72-36x^{2}-360x=0$$

Combine 72x and -360x to get -288x.

$$-288x+720-x\times 72-36x^{2}=0$$

Subtract 720 from both sides. Anything subtracted from zero gives its negation.

$$-288x-x\times 72-36x^{2}=-720$$

Multiply -1 and 72 to get -72.

$$-288x-72x-36x^{2}=-720$$

Combine -288x and -72x to get -360x.

$$-360x-36x^{2}=-720$$

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.

$$-36x^{2}-360x=-720$$

Divide both sides by -36.

$$\frac{-36x^{2}-360x}{-36}=-\frac{720}{-36}$$

Dividing by -36 undoes the multiplication by -36.

$$x^{2}+\left(-\frac{360}{-36}\right)x=-\frac{720}{-36}$$

Divide -360 by -36.

$$x^{2}+10x=-\frac{720}{-36}$$

Divide -720 by -36.

$$x^{2}+10x=20$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=20+5^{2}$$

Square 5.

$$x^{2}+10x+25=20+25$$

Add 20 to 25.

$$x^{2}+10x+25=45$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=45$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{45}$$

Simplify.

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

Subtract 5 from both sides of the equation.

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