Solve 1/x+1/x+10=16/75

$$\frac { 1 } { x } + \frac { 1 } { x + 10 } = \frac { 16 } { 75 }$$

Solve for x

$$x = \frac{5 \sqrt{481} - 5}{16} \approx 6.541160062$$

$$x=\frac{-5\sqrt{481}-5}{16}\approx -7.166160062$$

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 75x\left(x+10\right), the least common multiple of x,x+10,75.

$$75x+750+75x=16x\left(x+10\right)$$

Combine 75x and 75x to get 150x.

$$150x+750=16x\left(x+10\right)$$

Use the distributive property to multiply 16x by x+10.

$$150x+750=16x^{2}+160x$$

Subtract 16x^{2} from both sides.

$$150x+750-16x^{2}=160x$$

Subtract 160x from both sides.

$$150x+750-16x^{2}-160x=0$$

Combine 150x and -160x to get -10x.

$$-10x+750-16x^{2}=0$$

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.

$$-16x^{2}-10x+750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, -10 for b, and 750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-16\right)\times 750}}{2\left(-16\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-16\right)\times 750}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-\left(-10\right)±\sqrt{100+64\times 750}}{2\left(-16\right)}$$

Multiply 64 times 750.

$$x=\frac{-\left(-10\right)±\sqrt{100+48000}}{2\left(-16\right)}$$

Add 100 to 48000.

$$x=\frac{-\left(-10\right)±\sqrt{48100}}{2\left(-16\right)}$$

Take the square root of 48100.

$$x=\frac{-\left(-10\right)±10\sqrt{481}}{2\left(-16\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±10\sqrt{481}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{10±10\sqrt{481}}{-32}$$

Now solve the equation x=\frac{10±10\sqrt{481}}{-32} when ± is plus. Add 10 to 10\sqrt{481}.

$$x=\frac{10\sqrt{481}+10}{-32}$$

Divide 10+10\sqrt{481} by -32.

$$x=\frac{-5\sqrt{481}-5}{16}$$

Now solve the equation x=\frac{10±10\sqrt{481}}{-32} when ± is minus. Subtract 10\sqrt{481} from 10.

$$x=\frac{10-10\sqrt{481}}{-32}$$

Divide 10-10\sqrt{481} by -32.

$$x=\frac{5\sqrt{481}-5}{16}$$

The equation is now solved.

$$x=\frac{-5\sqrt{481}-5}{16} x=\frac{5\sqrt{481}-5}{16}$$

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 75x\left(x+10\right), the least common multiple of x,x+10,75.

$$75x+750+75x=16x\left(x+10\right)$$

Combine 75x and 75x to get 150x.

$$150x+750=16x\left(x+10\right)$$

Use the distributive property to multiply 16x by x+10.

$$150x+750=16x^{2}+160x$$

Subtract 16x^{2} from both sides.

$$150x+750-16x^{2}=160x$$

Subtract 160x from both sides.

$$150x+750-16x^{2}-160x=0$$

Combine 150x and -160x to get -10x.

$$-10x+750-16x^{2}=0$$

Subtract 750 from both sides. Anything subtracted from zero gives its negation.

$$-10x-16x^{2}=-750$$

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.

$$-16x^{2}-10x=-750$$

Divide both sides by -16.

$$\frac{-16x^{2}-10x}{-16}=-\frac{750}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\left(-\frac{10}{-16}\right)x=-\frac{750}{-16}$$

Reduce the fraction \frac{-10}{-16} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{8}x=-\frac{750}{-16}$$

Reduce the fraction \frac{-750}{-16} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{8}x=\frac{375}{8}$$

Divide \frac{5}{8}, the coefficient of the x term, by 2 to get \frac{5}{16}. Then add the square of \frac{5}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{8}x+\left(\frac{5}{16}\right)^{2}=\frac{375}{8}+\left(\frac{5}{16}\right)^{2}$$

Square \frac{5}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{8}x+\frac{25}{256}=\frac{375}{8}+\frac{25}{256}$$

Add \frac{375}{8} to \frac{25}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{8}x+\frac{25}{256}=\frac{12025}{256}$$

Factor x^{2}+\frac{5}{8}x+\frac{25}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{16}\right)^{2}=\frac{12025}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{16}\right)^{2}}=\sqrt{\frac{12025}{256}}$$

Simplify.

$$x+\frac{5}{16}=\frac{5\sqrt{481}}{16} x+\frac{5}{16}=-\frac{5\sqrt{481}}{16}$$

Subtract \frac{5}{16} from both sides of the equation.

$$x=\frac{5\sqrt{481}-5}{16} x=\frac{-5\sqrt{481}-5}{16}$$