Solve 60/x-40/13-x=3

$$\frac { 60 } { x } - \frac { 40 } { 13 - x } = 3$$

Solve for x

$$x = \frac{\sqrt{9961} + 139}{6} \approx 39.800801584$$

$$x = \frac{139 - \sqrt{9961}}{6} \approx 6.532531749$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,13 since division by zero is not defined. Multiply both sides of the equation by x\left(x-13\right), the least common multiple of x,13-x.

$$\left(x-13\right)\times 60-\left(-x\times 40\right)=3x\left(x-13\right)$$

Use the distributive property to multiply x-13 by 60.

$$60x-780-\left(-x\times 40\right)=3x\left(x-13\right)$$

Multiply -1 and 40 to get -40.

$$60x-780-\left(-40x\right)=3x\left(x-13\right)$$

The opposite of -40x is 40x.

$$60x-780+40x=3x\left(x-13\right)$$

Combine 60x and 40x to get 100x.

$$100x-780=3x\left(x-13\right)$$

Use the distributive property to multiply 3x by x-13.

$$100x-780=3x^{2}-39x$$

Subtract 3x^{2} from both sides.

$$100x-780-3x^{2}=-39x$$

Add 39x to both sides.

$$100x-780-3x^{2}+39x=0$$

Combine 100x and 39x to get 139x.

$$139x-780-3x^{2}=0$$

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.

$$-3x^{2}+139x-780=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 139 for b, and -780 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-139±\sqrt{139^{2}-4\left(-3\right)\left(-780\right)}}{2\left(-3\right)}$$

Square 139.

$$x=\frac{-139±\sqrt{19321-4\left(-3\right)\left(-780\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-139±\sqrt{19321+12\left(-780\right)}}{2\left(-3\right)}$$

Multiply 12 times -780.

$$x=\frac{-139±\sqrt{19321-9360}}{2\left(-3\right)}$$

Add 19321 to -9360.

$$x=\frac{-139±\sqrt{9961}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-139±\sqrt{9961}}{-6}$$

Now solve the equation x=\frac{-139±\sqrt{9961}}{-6} when ± is plus. Add -139 to \sqrt{9961}.

$$x=\frac{\sqrt{9961}-139}{-6}$$

Divide -139+\sqrt{9961} by -6.

$$x=\frac{139-\sqrt{9961}}{6}$$

Now solve the equation x=\frac{-139±\sqrt{9961}}{-6} when ± is minus. Subtract \sqrt{9961} from -139.

$$x=\frac{-\sqrt{9961}-139}{-6}$$

Divide -139-\sqrt{9961} by -6.

$$x=\frac{\sqrt{9961}+139}{6}$$

The equation is now solved.

$$x=\frac{139-\sqrt{9961}}{6} x=\frac{\sqrt{9961}+139}{6}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,13 since division by zero is not defined. Multiply both sides of the equation by x\left(x-13\right), the least common multiple of x,13-x.

$$\left(x-13\right)\times 60-\left(-x\times 40\right)=3x\left(x-13\right)$$

Use the distributive property to multiply x-13 by 60.

$$60x-780-\left(-x\times 40\right)=3x\left(x-13\right)$$

Multiply -1 and 40 to get -40.

$$60x-780-\left(-40x\right)=3x\left(x-13\right)$$

The opposite of -40x is 40x.

$$60x-780+40x=3x\left(x-13\right)$$

Combine 60x and 40x to get 100x.

$$100x-780=3x\left(x-13\right)$$

Use the distributive property to multiply 3x by x-13.

$$100x-780=3x^{2}-39x$$

Subtract 3x^{2} from both sides.

$$100x-780-3x^{2}=-39x$$

Add 39x to both sides.

$$100x-780-3x^{2}+39x=0$$

Combine 100x and 39x to get 139x.

$$139x-780-3x^{2}=0$$

Add 780 to both sides. Anything plus zero gives itself.

$$139x-3x^{2}=780$$

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.

$$-3x^{2}+139x=780$$

Divide both sides by -3.

$$\frac{-3x^{2}+139x}{-3}=\frac{780}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{139}{-3}x=\frac{780}{-3}$$

Divide 139 by -3.

$$x^{2}-\frac{139}{3}x=\frac{780}{-3}$$

Divide 780 by -3.

$$x^{2}-\frac{139}{3}x=-260$$

Divide -\frac{139}{3}, the coefficient of the x term, by 2 to get -\frac{139}{6}. Then add the square of -\frac{139}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{139}{3}x+\left(-\frac{139}{6}\right)^{2}=-260+\left(-\frac{139}{6}\right)^{2}$$

Square -\frac{139}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{139}{3}x+\frac{19321}{36}=-260+\frac{19321}{36}$$

Add -260 to \frac{19321}{36}.

$$x^{2}-\frac{139}{3}x+\frac{19321}{36}=\frac{9961}{36}$$

Factor x^{2}-\frac{139}{3}x+\frac{19321}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{139}{6}\right)^{2}=\frac{9961}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{139}{6}\right)^{2}}=\sqrt{\frac{9961}{36}}$$

Simplify.

$$x-\frac{139}{6}=\frac{\sqrt{9961}}{6} x-\frac{139}{6}=-\frac{\sqrt{9961}}{6}$$

Add \frac{139}{6} to both sides of the equation.

$$x=\frac{\sqrt{9961}+139}{6} x=\frac{139-\sqrt{9961}}{6}$$