Solve 94+240/x=120/10+120/x+10

$$94 + \frac { 240 } { x } = \frac { 120 } { 10 } + \frac { 120 } { x + 10 }$$

Solve for x

$$x=\frac{5\sqrt{241}-235}{41}\approx -3.838515281$$

$$x=\frac{-5\sqrt{241}-235}{41}\approx -7.624899353$$

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

$$10x\left(x+10\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

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

$$\left(10x^{2}+100x\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Use the distributive property to multiply 10x^{2}+100x by 94.

$$940x^{2}+9400x+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Use the distributive property to multiply 10x+100 by 240.

$$940x^{2}+9400x+2400x+24000=x\left(x+10\right)\times 120+10x\times 120$$

Combine 9400x and 2400x to get 11800x.

$$940x^{2}+11800x+24000=x\left(x+10\right)\times 120+10x\times 120$$

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

$$940x^{2}+11800x+24000=\left(x^{2}+10x\right)\times 120+10x\times 120$$

Use the distributive property to multiply x^{2}+10x by 120.

$$940x^{2}+11800x+24000=120x^{2}+1200x+10x\times 120$$

Multiply 10 and 120 to get 1200.

$$940x^{2}+11800x+24000=120x^{2}+1200x+1200x$$

Combine 1200x and 1200x to get 2400x.

$$940x^{2}+11800x+24000=120x^{2}+2400x$$

Subtract 120x^{2} from both sides.

$$940x^{2}+11800x+24000-120x^{2}=2400x$$

Combine 940x^{2} and -120x^{2} to get 820x^{2}.

$$820x^{2}+11800x+24000=2400x$$

Subtract 2400x from both sides.

$$820x^{2}+11800x+24000-2400x=0$$

Combine 11800x and -2400x to get 9400x.

$$820x^{2}+9400x+24000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 820 for a, 9400 for b, and 24000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9400±\sqrt{9400^{2}-4\times 820\times 24000}}{2\times 820}$$

Square 9400.

$$x=\frac{-9400±\sqrt{88360000-4\times 820\times 24000}}{2\times 820}$$

Multiply -4 times 820.

$$x=\frac{-9400±\sqrt{88360000-3280\times 24000}}{2\times 820}$$

Multiply -3280 times 24000.

$$x=\frac{-9400±\sqrt{88360000-78720000}}{2\times 820}$$

Add 88360000 to -78720000.

$$x=\frac{-9400±\sqrt{9640000}}{2\times 820}$$

Take the square root of 9640000.

$$x=\frac{-9400±200\sqrt{241}}{2\times 820}$$

Multiply 2 times 820.

$$x=\frac{-9400±200\sqrt{241}}{1640}$$

Now solve the equation x=\frac{-9400±200\sqrt{241}}{1640} when ± is plus. Add -9400 to 200\sqrt{241}.

$$x=\frac{200\sqrt{241}-9400}{1640}$$

Divide -9400+200\sqrt{241} by 1640.

$$x=\frac{5\sqrt{241}-235}{41}$$

Now solve the equation x=\frac{-9400±200\sqrt{241}}{1640} when ± is minus. Subtract 200\sqrt{241} from -9400.

$$x=\frac{-200\sqrt{241}-9400}{1640}$$

Divide -9400-200\sqrt{241} by 1640.

$$x=\frac{-5\sqrt{241}-235}{41}$$

The equation is now solved.

$$x=\frac{5\sqrt{241}-235}{41} x=\frac{-5\sqrt{241}-235}{41}$$

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

$$10x\left(x+10\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

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

$$\left(10x^{2}+100x\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Use the distributive property to multiply 10x^{2}+100x by 94.

$$940x^{2}+9400x+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Use the distributive property to multiply 10x+100 by 240.

$$940x^{2}+9400x+2400x+24000=x\left(x+10\right)\times 120+10x\times 120$$

Combine 9400x and 2400x to get 11800x.

$$940x^{2}+11800x+24000=x\left(x+10\right)\times 120+10x\times 120$$

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

$$940x^{2}+11800x+24000=\left(x^{2}+10x\right)\times 120+10x\times 120$$

Use the distributive property to multiply x^{2}+10x by 120.

$$940x^{2}+11800x+24000=120x^{2}+1200x+10x\times 120$$

Multiply 10 and 120 to get 1200.

$$940x^{2}+11800x+24000=120x^{2}+1200x+1200x$$

Combine 1200x and 1200x to get 2400x.

$$940x^{2}+11800x+24000=120x^{2}+2400x$$

Subtract 120x^{2} from both sides.

$$940x^{2}+11800x+24000-120x^{2}=2400x$$

Combine 940x^{2} and -120x^{2} to get 820x^{2}.

$$820x^{2}+11800x+24000=2400x$$

Subtract 2400x from both sides.

$$820x^{2}+11800x+24000-2400x=0$$

Combine 11800x and -2400x to get 9400x.

$$820x^{2}+9400x+24000=0$$

Subtract 24000 from both sides. Anything subtracted from zero gives its negation.

$$820x^{2}+9400x=-24000$$

Divide both sides by 820.

$$\frac{820x^{2}+9400x}{820}=-\frac{24000}{820}$$

Dividing by 820 undoes the multiplication by 820.

$$x^{2}+\frac{9400}{820}x=-\frac{24000}{820}$$

Reduce the fraction \frac{9400}{820} to lowest terms by extracting and canceling out 20.

$$x^{2}+\frac{470}{41}x=-\frac{24000}{820}$$

Reduce the fraction \frac{-24000}{820} to lowest terms by extracting and canceling out 20.

$$x^{2}+\frac{470}{41}x=-\frac{1200}{41}$$

Divide \frac{470}{41}, the coefficient of the x term, by 2 to get \frac{235}{41}. Then add the square of \frac{235}{41} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{470}{41}x+\left(\frac{235}{41}\right)^{2}=-\frac{1200}{41}+\left(\frac{235}{41}\right)^{2}$$

Square \frac{235}{41} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{470}{41}x+\frac{55225}{1681}=-\frac{1200}{41}+\frac{55225}{1681}$$

Add -\frac{1200}{41} to \frac{55225}{1681} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{470}{41}x+\frac{55225}{1681}=\frac{6025}{1681}$$

Factor x^{2}+\frac{470}{41}x+\frac{55225}{1681}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{235}{41}\right)^{2}=\frac{6025}{1681}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{235}{41}\right)^{2}}=\sqrt{\frac{6025}{1681}}$$

Simplify.

$$x+\frac{235}{41}=\frac{5\sqrt{241}}{41} x+\frac{235}{41}=-\frac{5\sqrt{241}}{41}$$

Subtract \frac{235}{41} from both sides of the equation.

$$x=\frac{5\sqrt{241}-235}{41} x=\frac{-5\sqrt{241}-235}{41}$$