Solve 84/x+84/x+10=7/2

$$\frac{ 84 }{ x } + \frac{ 84 }{ x+10 } = \frac{ 7 }{ 2 }$$

Solve for x

$$x=\sqrt{601}+19\approx 43.515301344$$

$$x=19-\sqrt{601}\approx -5.515301344$$

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 2x\left(x+10\right), the least common multiple of x,x+10,2.

$$\left(2x+20\right)\times 84+2x\times 84=7x\left(x+10\right)$$

Use the distributive property to multiply 2x+20 by 84.

$$168x+1680+2x\times 84=7x\left(x+10\right)$$

Multiply 2 and 84 to get 168.

$$168x+1680+168x=7x\left(x+10\right)$$

Combine 168x and 168x to get 336x.

$$336x+1680=7x\left(x+10\right)$$

Use the distributive property to multiply 7x by x+10.

$$336x+1680=7x^{2}+70x$$

Subtract 7x^{2} from both sides.

$$336x+1680-7x^{2}=70x$$

Subtract 70x from both sides.

$$336x+1680-7x^{2}-70x=0$$

Combine 336x and -70x to get 266x.

$$266x+1680-7x^{2}=0$$

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.

$$-7x^{2}+266x+1680=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 266 for b, and 1680 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-266±\sqrt{266^{2}-4\left(-7\right)\times 1680}}{2\left(-7\right)}$$

Square 266.

$$x=\frac{-266±\sqrt{70756-4\left(-7\right)\times 1680}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-266±\sqrt{70756+28\times 1680}}{2\left(-7\right)}$$

Multiply 28 times 1680.

$$x=\frac{-266±\sqrt{70756+47040}}{2\left(-7\right)}$$

Add 70756 to 47040.

$$x=\frac{-266±\sqrt{117796}}{2\left(-7\right)}$$

Take the square root of 117796.

$$x=\frac{-266±14\sqrt{601}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-266±14\sqrt{601}}{-14}$$

Now solve the equation x=\frac{-266±14\sqrt{601}}{-14} when ± is plus. Add -266 to 14\sqrt{601}.

$$x=\frac{14\sqrt{601}-266}{-14}$$

Divide -266+14\sqrt{601} by -14.

$$x=19-\sqrt{601}$$

Now solve the equation x=\frac{-266±14\sqrt{601}}{-14} when ± is minus. Subtract 14\sqrt{601} from -266.

$$x=\frac{-14\sqrt{601}-266}{-14}$$

Divide -266-14\sqrt{601} by -14.

$$x=\sqrt{601}+19$$

The equation is now solved.

$$x=19-\sqrt{601} x=\sqrt{601}+19$$

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 2x\left(x+10\right), the least common multiple of x,x+10,2.

$$\left(2x+20\right)\times 84+2x\times 84=7x\left(x+10\right)$$

Use the distributive property to multiply 2x+20 by 84.

$$168x+1680+2x\times 84=7x\left(x+10\right)$$

Multiply 2 and 84 to get 168.

$$168x+1680+168x=7x\left(x+10\right)$$

Combine 168x and 168x to get 336x.

$$336x+1680=7x\left(x+10\right)$$

Use the distributive property to multiply 7x by x+10.

$$336x+1680=7x^{2}+70x$$

Subtract 7x^{2} from both sides.

$$336x+1680-7x^{2}=70x$$

Subtract 70x from both sides.

$$336x+1680-7x^{2}-70x=0$$

Combine 336x and -70x to get 266x.

$$266x+1680-7x^{2}=0$$

Subtract 1680 from both sides. Anything subtracted from zero gives its negation.

$$266x-7x^{2}=-1680$$

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.

$$-7x^{2}+266x=-1680$$

Divide both sides by -7.

$$\frac{-7x^{2}+266x}{-7}=-\frac{1680}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{266}{-7}x=-\frac{1680}{-7}$$

Divide 266 by -7.

$$x^{2}-38x=-\frac{1680}{-7}$$

Divide -1680 by -7.

$$x^{2}-38x=240$$

Divide -38, the coefficient of the x term, by 2 to get -19. Then add the square of -19 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-38x+\left(-19\right)^{2}=240+\left(-19\right)^{2}$$

Square -19.

$$x^{2}-38x+361=240+361$$

Add 240 to 361.

$$x^{2}-38x+361=601$$

Factor x^{2}-38x+361. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-19\right)^{2}=601$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-19\right)^{2}}=\sqrt{601}$$

Simplify.

$$x-19=\sqrt{601} x-19=-\sqrt{601}$$

Add 19 to both sides of the equation.

$$x=\sqrt{601}+19 x=19-\sqrt{601}$$