Solve 150/x+45/x+10=315/60

$$\frac{ 150 }{ x } + \frac{ 45 }{ x+10 } = 3 \frac { 15 } { 60 }$$

Solve for x

$$x=\frac{5\sqrt{7345}}{13}+25\approx 57.962682863$$

$$x=-\frac{5\sqrt{7345}}{13}+25\approx -7.962682863$$

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 60x\left(x+10\right), the least common multiple of x,x+10,60.

$$\left(60x+600\right)\times 150+60x\times 45=x\left(x+10\right)\left(3\times 60+15\right)$$

Use the distributive property to multiply 60x+600 by 150.

$$9000x+90000+60x\times 45=x\left(x+10\right)\left(3\times 60+15\right)$$

Multiply 60 and 45 to get 2700.

$$9000x+90000+2700x=x\left(x+10\right)\left(3\times 60+15\right)$$

Combine 9000x and 2700x to get 11700x.

$$11700x+90000=x\left(x+10\right)\left(3\times 60+15\right)$$

Multiply 3 and 60 to get 180.

$$11700x+90000=x\left(x+10\right)\left(180+15\right)$$

Add 180 and 15 to get 195.

$$11700x+90000=x\left(x+10\right)\times 195$$

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

$$11700x+90000=\left(x^{2}+10x\right)\times 195$$

Use the distributive property to multiply x^{2}+10x by 195.

$$11700x+90000=195x^{2}+1950x$$

Subtract 195x^{2} from both sides.

$$11700x+90000-195x^{2}=1950x$$

Subtract 1950x from both sides.

$$11700x+90000-195x^{2}-1950x=0$$

Combine 11700x and -1950x to get 9750x.

$$9750x+90000-195x^{2}=0$$

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.

$$-195x^{2}+9750x+90000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -195 for a, 9750 for b, and 90000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9750±\sqrt{9750^{2}-4\left(-195\right)\times 90000}}{2\left(-195\right)}$$

Square 9750.

$$x=\frac{-9750±\sqrt{95062500-4\left(-195\right)\times 90000}}{2\left(-195\right)}$$

Multiply -4 times -195.

$$x=\frac{-9750±\sqrt{95062500+780\times 90000}}{2\left(-195\right)}$$

Multiply 780 times 90000.

$$x=\frac{-9750±\sqrt{95062500+70200000}}{2\left(-195\right)}$$

Add 95062500 to 70200000.

$$x=\frac{-9750±\sqrt{165262500}}{2\left(-195\right)}$$

Take the square root of 165262500.

$$x=\frac{-9750±150\sqrt{7345}}{2\left(-195\right)}$$

Multiply 2 times -195.

$$x=\frac{-9750±150\sqrt{7345}}{-390}$$

Now solve the equation x=\frac{-9750±150\sqrt{7345}}{-390} when ± is plus. Add -9750 to 150\sqrt{7345}.

$$x=\frac{150\sqrt{7345}-9750}{-390}$$

Divide -9750+150\sqrt{7345} by -390.

$$x=-\frac{5\sqrt{7345}}{13}+25$$

Now solve the equation x=\frac{-9750±150\sqrt{7345}}{-390} when ± is minus. Subtract 150\sqrt{7345} from -9750.

$$x=\frac{-150\sqrt{7345}-9750}{-390}$$

Divide -9750-150\sqrt{7345} by -390.

$$x=\frac{5\sqrt{7345}}{13}+25$$

The equation is now solved.

$$x=-\frac{5\sqrt{7345}}{13}+25 x=\frac{5\sqrt{7345}}{13}+25$$

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 60x\left(x+10\right), the least common multiple of x,x+10,60.

$$\left(60x+600\right)\times 150+60x\times 45=x\left(x+10\right)\left(3\times 60+15\right)$$

Use the distributive property to multiply 60x+600 by 150.

$$9000x+90000+60x\times 45=x\left(x+10\right)\left(3\times 60+15\right)$$

Multiply 60 and 45 to get 2700.

$$9000x+90000+2700x=x\left(x+10\right)\left(3\times 60+15\right)$$

Combine 9000x and 2700x to get 11700x.

$$11700x+90000=x\left(x+10\right)\left(3\times 60+15\right)$$

Multiply 3 and 60 to get 180.

$$11700x+90000=x\left(x+10\right)\left(180+15\right)$$

Add 180 and 15 to get 195.

$$11700x+90000=x\left(x+10\right)\times 195$$

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

$$11700x+90000=\left(x^{2}+10x\right)\times 195$$

Use the distributive property to multiply x^{2}+10x by 195.

$$11700x+90000=195x^{2}+1950x$$

Subtract 195x^{2} from both sides.

$$11700x+90000-195x^{2}=1950x$$

Subtract 1950x from both sides.

$$11700x+90000-195x^{2}-1950x=0$$

Combine 11700x and -1950x to get 9750x.

$$9750x+90000-195x^{2}=0$$

Subtract 90000 from both sides. Anything subtracted from zero gives its negation.

$$9750x-195x^{2}=-90000$$

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.

$$-195x^{2}+9750x=-90000$$

Divide both sides by -195.

$$\frac{-195x^{2}+9750x}{-195}=-\frac{90000}{-195}$$

Dividing by -195 undoes the multiplication by -195.

$$x^{2}+\frac{9750}{-195}x=-\frac{90000}{-195}$$

Divide 9750 by -195.

$$x^{2}-50x=-\frac{90000}{-195}$$

Reduce the fraction \frac{-90000}{-195} to lowest terms by extracting and canceling out 15.

$$x^{2}-50x=\frac{6000}{13}$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=\frac{6000}{13}+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=\frac{6000}{13}+625$$

Add \frac{6000}{13} to 625.

$$x^{2}-50x+625=\frac{14125}{13}$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=\frac{14125}{13}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{\frac{14125}{13}}$$

Simplify.

$$x-25=\frac{5\sqrt{7345}}{13} x-25=-\frac{5\sqrt{7345}}{13}$$

Add 25 to both sides of the equation.

$$x=\frac{5\sqrt{7345}}{13}+25 x=-\frac{5\sqrt{7345}}{13}+25$$