Solve 180/x+180/x-5+3/2=10

$$\frac{ 180 }{ x } + \frac{ 180 }{ x-5 } + \frac{ 3 }{ 2 } = 10$$

Solve for x

$$x=45$$

$$x = \frac{40}{17} = 2\frac{6}{17} \approx 2.352941176$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-5\right), the least common multiple of x,x-5,2.

$$\left(2x-10\right)\times 180+2x\times 180+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Use the distributive property to multiply 2x-10 by 180.

$$360x-1800+2x\times 180+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Multiply 2 and 180 to get 360.

$$360x-1800+360x+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Combine 360x and 360x to get 720x.

$$720x-1800+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Multiply 2 and \frac{3}{2} to get 3.

$$720x-1800+3x\left(x-5\right)=20x\left(x-5\right)$$

Use the distributive property to multiply 3x by x-5.

$$720x-1800+3x^{2}-15x=20x\left(x-5\right)$$

Combine 720x and -15x to get 705x.

$$705x-1800+3x^{2}=20x\left(x-5\right)$$

Use the distributive property to multiply 20x by x-5.

$$705x-1800+3x^{2}=20x^{2}-100x$$

Subtract 20x^{2} from both sides.

$$705x-1800+3x^{2}-20x^{2}=-100x$$

Combine 3x^{2} and -20x^{2} to get -17x^{2}.

$$705x-1800-17x^{2}=-100x$$

Add 100x to both sides.

$$705x-1800-17x^{2}+100x=0$$

Combine 705x and 100x to get 805x.

$$805x-1800-17x^{2}=0$$

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.

$$-17x^{2}+805x-1800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -17 for a, 805 for b, and -1800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-805±\sqrt{805^{2}-4\left(-17\right)\left(-1800\right)}}{2\left(-17\right)}$$

Square 805.

$$x=\frac{-805±\sqrt{648025-4\left(-17\right)\left(-1800\right)}}{2\left(-17\right)}$$

Multiply -4 times -17.

$$x=\frac{-805±\sqrt{648025+68\left(-1800\right)}}{2\left(-17\right)}$$

Multiply 68 times -1800.

$$x=\frac{-805±\sqrt{648025-122400}}{2\left(-17\right)}$$

Add 648025 to -122400.

$$x=\frac{-805±\sqrt{525625}}{2\left(-17\right)}$$

Take the square root of 525625.

$$x=\frac{-805±725}{2\left(-17\right)}$$

Multiply 2 times -17.

$$x=\frac{-805±725}{-34}$$

Now solve the equation x=\frac{-805±725}{-34} when ± is plus. Add -805 to 725.

$$x=-\frac{80}{-34}$$

Reduce the fraction \frac{-80}{-34} to lowest terms by extracting and canceling out 2.

$$x=\frac{40}{17}$$

Now solve the equation x=\frac{-805±725}{-34} when ± is minus. Subtract 725 from -805.

$$x=-\frac{1530}{-34}$$

Divide -1530 by -34.

$$x=45$$

The equation is now solved.

$$x=\frac{40}{17} x=45$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-5\right), the least common multiple of x,x-5,2.

$$\left(2x-10\right)\times 180+2x\times 180+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Use the distributive property to multiply 2x-10 by 180.

$$360x-1800+2x\times 180+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Multiply 2 and 180 to get 360.

$$360x-1800+360x+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Combine 360x and 360x to get 720x.

$$720x-1800+2x\left(x-5\right)\times \frac{3}{2}=20x\left(x-5\right)$$

Multiply 2 and \frac{3}{2} to get 3.

$$720x-1800+3x\left(x-5\right)=20x\left(x-5\right)$$

Use the distributive property to multiply 3x by x-5.

$$720x-1800+3x^{2}-15x=20x\left(x-5\right)$$

Combine 720x and -15x to get 705x.

$$705x-1800+3x^{2}=20x\left(x-5\right)$$

Use the distributive property to multiply 20x by x-5.

$$705x-1800+3x^{2}=20x^{2}-100x$$

Subtract 20x^{2} from both sides.

$$705x-1800+3x^{2}-20x^{2}=-100x$$

Combine 3x^{2} and -20x^{2} to get -17x^{2}.

$$705x-1800-17x^{2}=-100x$$

Add 100x to both sides.

$$705x-1800-17x^{2}+100x=0$$

Combine 705x and 100x to get 805x.

$$805x-1800-17x^{2}=0$$

Add 1800 to both sides. Anything plus zero gives itself.

$$805x-17x^{2}=1800$$

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.

$$-17x^{2}+805x=1800$$

Divide both sides by -17.

$$\frac{-17x^{2}+805x}{-17}=\frac{1800}{-17}$$

Dividing by -17 undoes the multiplication by -17.

$$x^{2}+\frac{805}{-17}x=\frac{1800}{-17}$$

Divide 805 by -17.

$$x^{2}-\frac{805}{17}x=\frac{1800}{-17}$$

Divide 1800 by -17.

$$x^{2}-\frac{805}{17}x=-\frac{1800}{17}$$

Divide -\frac{805}{17}, the coefficient of the x term, by 2 to get -\frac{805}{34}. Then add the square of -\frac{805}{34} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{805}{17}x+\left(-\frac{805}{34}\right)^{2}=-\frac{1800}{17}+\left(-\frac{805}{34}\right)^{2}$$

Square -\frac{805}{34} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{805}{17}x+\frac{648025}{1156}=-\frac{1800}{17}+\frac{648025}{1156}$$

Add -\frac{1800}{17} to \frac{648025}{1156} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{805}{17}x+\frac{648025}{1156}=\frac{525625}{1156}$$

Factor x^{2}-\frac{805}{17}x+\frac{648025}{1156}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{805}{34}\right)^{2}=\frac{525625}{1156}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{805}{34}\right)^{2}}=\sqrt{\frac{525625}{1156}}$$

Simplify.

$$x-\frac{805}{34}=\frac{725}{34} x-\frac{805}{34}=-\frac{725}{34}$$

Add \frac{805}{34} to both sides of the equation.

$$x=45 x=\frac{40}{17}$$