Solve 6/x+10+3=50/x

$$\frac { 6 } { x + 10 } + 3 = \frac { 50 } { x }$$

Solve for x

$$x = \frac{\sqrt{1549} + 7}{3} \approx 15.452445769$$

$$x=\frac{7-\sqrt{1549}}{3}\approx -10.785779103$$

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+10,x.

$$x\times 6+x\left(x+10\right)\times 3=\left(x+10\right)\times 50$$

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

$$x\times 6+\left(x^{2}+10x\right)\times 3=\left(x+10\right)\times 50$$

Use the distributive property to multiply x^{2}+10x by 3.

$$x\times 6+3x^{2}+30x=\left(x+10\right)\times 50$$

Combine x\times 6 and 30x to get 36x.

$$36x+3x^{2}=\left(x+10\right)\times 50$$

Use the distributive property to multiply x+10 by 50.

$$36x+3x^{2}=50x+500$$

Subtract 50x from both sides.

$$36x+3x^{2}-50x=500$$

Combine 36x and -50x to get -14x.

$$-14x+3x^{2}=500$$

Subtract 500 from both sides.

$$-14x+3x^{2}-500=0$$

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.

$$3x^{2}-14x-500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -14 for b, and -500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 3\left(-500\right)}}{2\times 3}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 3\left(-500\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-14\right)±\sqrt{196-12\left(-500\right)}}{2\times 3}$$

Multiply -12 times -500.

$$x=\frac{-\left(-14\right)±\sqrt{196+6000}}{2\times 3}$$

Add 196 to 6000.

$$x=\frac{-\left(-14\right)±\sqrt{6196}}{2\times 3}$$

Take the square root of 6196.

$$x=\frac{-\left(-14\right)±2\sqrt{1549}}{2\times 3}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{1549}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{14±2\sqrt{1549}}{6}$$

Now solve the equation x=\frac{14±2\sqrt{1549}}{6} when ± is plus. Add 14 to 2\sqrt{1549}.

$$x=\frac{2\sqrt{1549}+14}{6}$$

Divide 14+2\sqrt{1549} by 6.

$$x=\frac{\sqrt{1549}+7}{3}$$

Now solve the equation x=\frac{14±2\sqrt{1549}}{6} when ± is minus. Subtract 2\sqrt{1549} from 14.

$$x=\frac{14-2\sqrt{1549}}{6}$$

Divide 14-2\sqrt{1549} by 6.

$$x=\frac{7-\sqrt{1549}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{1549}+7}{3} x=\frac{7-\sqrt{1549}}{3}$$

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+10,x.

$$x\times 6+x\left(x+10\right)\times 3=\left(x+10\right)\times 50$$

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

$$x\times 6+\left(x^{2}+10x\right)\times 3=\left(x+10\right)\times 50$$

Use the distributive property to multiply x^{2}+10x by 3.

$$x\times 6+3x^{2}+30x=\left(x+10\right)\times 50$$

Combine x\times 6 and 30x to get 36x.

$$36x+3x^{2}=\left(x+10\right)\times 50$$

Use the distributive property to multiply x+10 by 50.

$$36x+3x^{2}=50x+500$$

Subtract 50x from both sides.

$$36x+3x^{2}-50x=500$$

Combine 36x and -50x to get -14x.

$$-14x+3x^{2}=500$$

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.

$$3x^{2}-14x=500$$

Divide both sides by 3.

$$\frac{3x^{2}-14x}{3}=\frac{500}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{14}{3}x=\frac{500}{3}$$

Divide -\frac{14}{3}, the coefficient of the x term, by 2 to get -\frac{7}{3}. Then add the square of -\frac{7}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=\frac{500}{3}+\left(-\frac{7}{3}\right)^{2}$$

Square -\frac{7}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{500}{3}+\frac{49}{9}$$

Add \frac{500}{3} to \frac{49}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{1549}{9}$$

Factor x^{2}-\frac{14}{3}x+\frac{49}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{3}\right)^{2}=\frac{1549}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{\frac{1549}{9}}$$

Simplify.

$$x-\frac{7}{3}=\frac{\sqrt{1549}}{3} x-\frac{7}{3}=-\frac{\sqrt{1549}}{3}$$

Add \frac{7}{3} to both sides of the equation.

$$x=\frac{\sqrt{1549}+7}{3} x=\frac{7-\sqrt{1549}}{3}$$