Solve 1/x+1/x+3=40/13

$$\frac{ 1 }{ x } + \frac{ 1 }{ x+3 } = \frac{ 40 }{ 13 }$$

Solve for x

$$x=\frac{\sqrt{3769}-47}{40}\approx 0.359804548$$

$$x=\frac{-\sqrt{3769}-47}{40}\approx -2.709804548$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 13x\left(x+3\right), the least common multiple of x,x+3,13.

$$13x+39+13x=40x\left(x+3\right)$$

Combine 13x and 13x to get 26x.

$$26x+39=40x\left(x+3\right)$$

Use the distributive property to multiply 40x by x+3.

$$26x+39=40x^{2}+120x$$

Subtract 40x^{2} from both sides.

$$26x+39-40x^{2}=120x$$

Subtract 120x from both sides.

$$26x+39-40x^{2}-120x=0$$

Combine 26x and -120x to get -94x.

$$-94x+39-40x^{2}=0$$

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.

$$-40x^{2}-94x+39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -40 for a, -94 for b, and 39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-94\right)±\sqrt{\left(-94\right)^{2}-4\left(-40\right)\times 39}}{2\left(-40\right)}$$

Square -94.

$$x=\frac{-\left(-94\right)±\sqrt{8836-4\left(-40\right)\times 39}}{2\left(-40\right)}$$

Multiply -4 times -40.

$$x=\frac{-\left(-94\right)±\sqrt{8836+160\times 39}}{2\left(-40\right)}$$

Multiply 160 times 39.

$$x=\frac{-\left(-94\right)±\sqrt{8836+6240}}{2\left(-40\right)}$$

Add 8836 to 6240.

$$x=\frac{-\left(-94\right)±\sqrt{15076}}{2\left(-40\right)}$$

Take the square root of 15076.

$$x=\frac{-\left(-94\right)±2\sqrt{3769}}{2\left(-40\right)}$$

The opposite of -94 is 94.

$$x=\frac{94±2\sqrt{3769}}{2\left(-40\right)}$$

Multiply 2 times -40.

$$x=\frac{94±2\sqrt{3769}}{-80}$$

Now solve the equation x=\frac{94±2\sqrt{3769}}{-80} when ± is plus. Add 94 to 2\sqrt{3769}.

$$x=\frac{2\sqrt{3769}+94}{-80}$$

Divide 94+2\sqrt{3769} by -80.

$$x=\frac{-\sqrt{3769}-47}{40}$$

Now solve the equation x=\frac{94±2\sqrt{3769}}{-80} when ± is minus. Subtract 2\sqrt{3769} from 94.

$$x=\frac{94-2\sqrt{3769}}{-80}$$

Divide 94-2\sqrt{3769} by -80.

$$x=\frac{\sqrt{3769}-47}{40}$$

The equation is now solved.

$$x=\frac{-\sqrt{3769}-47}{40} x=\frac{\sqrt{3769}-47}{40}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 13x\left(x+3\right), the least common multiple of x,x+3,13.

$$13x+39+13x=40x\left(x+3\right)$$

Combine 13x and 13x to get 26x.

$$26x+39=40x\left(x+3\right)$$

Use the distributive property to multiply 40x by x+3.

$$26x+39=40x^{2}+120x$$

Subtract 40x^{2} from both sides.

$$26x+39-40x^{2}=120x$$

Subtract 120x from both sides.

$$26x+39-40x^{2}-120x=0$$

Combine 26x and -120x to get -94x.

$$-94x+39-40x^{2}=0$$

Subtract 39 from both sides. Anything subtracted from zero gives its negation.

$$-94x-40x^{2}=-39$$

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.

$$-40x^{2}-94x=-39$$

Divide both sides by -40.

$$\frac{-40x^{2}-94x}{-40}=-\frac{39}{-40}$$

Dividing by -40 undoes the multiplication by -40.

$$x^{2}+\left(-\frac{94}{-40}\right)x=-\frac{39}{-40}$$

Reduce the fraction \frac{-94}{-40} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{47}{20}x=-\frac{39}{-40}$$

Divide -39 by -40.

$$x^{2}+\frac{47}{20}x=\frac{39}{40}$$

Divide \frac{47}{20}, the coefficient of the x term, by 2 to get \frac{47}{40}. Then add the square of \frac{47}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{47}{20}x+\left(\frac{47}{40}\right)^{2}=\frac{39}{40}+\left(\frac{47}{40}\right)^{2}$$

Square \frac{47}{40} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{47}{20}x+\frac{2209}{1600}=\frac{39}{40}+\frac{2209}{1600}$$

Add \frac{39}{40} to \frac{2209}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{47}{20}x+\frac{2209}{1600}=\frac{3769}{1600}$$

Factor x^{2}+\frac{47}{20}x+\frac{2209}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{47}{40}\right)^{2}=\frac{3769}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{47}{40}\right)^{2}}=\sqrt{\frac{3769}{1600}}$$

Simplify.

$$x+\frac{47}{40}=\frac{\sqrt{3769}}{40} x+\frac{47}{40}=-\frac{\sqrt{3769}}{40}$$

Subtract \frac{47}{40} from both sides of the equation.

$$x=\frac{\sqrt{3769}-47}{40} x=\frac{-\sqrt{3769}-47}{40}$$