Solve 1/x(16-9/x)=64/9

$$\frac { 1 } { x } ( 16 - \frac { 9 } { x } ) = \frac { 64 } { 9 }$$

Solve for x

$$x = \frac{9}{8} = 1\frac{1}{8} = 1.125$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 9x, the least common multiple of x,9.

$$9\times 1\left(16-\frac{9}{x}\right)=64x$$

Multiply 9 and 1 to get 9.

$$9\left(16-\frac{9}{x}\right)=64x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{x}{x}.

$$9\left(\frac{16x}{x}-\frac{9}{x}\right)=64x$$

Since \frac{16x}{x} and \frac{9}{x} have the same denominator, subtract them by subtracting their numerators.

$$9\times \frac{16x-9}{x}=64x$$

Express 9\times \frac{16x-9}{x} as a single fraction.

$$\frac{9\left(16x-9\right)}{x}=64x$$

Use the distributive property to multiply 9 by 16x-9.

$$\frac{144x-81}{x}=64x$$

Subtract 64x from both sides.

$$\frac{144x-81}{x}-64x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -64x times \frac{x}{x}.

$$\frac{144x-81}{x}+\frac{-64xx}{x}=0$$

Since \frac{144x-81}{x} and \frac{-64xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{144x-81-64xx}{x}=0$$

Do the multiplications in 144x-81-64xx.

$$\frac{144x-81-64x^{2}}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$144x-81-64x^{2}=0$$

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.

$$-64x^{2}+144x-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -64 for a, 144 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-144±\sqrt{144^{2}-4\left(-64\right)\left(-81\right)}}{2\left(-64\right)}$$

Square 144.

$$x=\frac{-144±\sqrt{20736-4\left(-64\right)\left(-81\right)}}{2\left(-64\right)}$$

Multiply -4 times -64.

$$x=\frac{-144±\sqrt{20736+256\left(-81\right)}}{2\left(-64\right)}$$

Multiply 256 times -81.

$$x=\frac{-144±\sqrt{20736-20736}}{2\left(-64\right)}$$

Add 20736 to -20736.

$$x=\frac{-144±\sqrt{0}}{2\left(-64\right)}$$

Take the square root of 0.

$$x=-\frac{144}{2\left(-64\right)}$$

Multiply 2 times -64.

$$x=-\frac{144}{-128}$$

Reduce the fraction \frac{-144}{-128} to lowest terms by extracting and canceling out 16.

$$x=\frac{9}{8}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 9x, the least common multiple of x,9.

$$9\times 1\left(16-\frac{9}{x}\right)=64x$$

Multiply 9 and 1 to get 9.

$$9\left(16-\frac{9}{x}\right)=64x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{x}{x}.

$$9\left(\frac{16x}{x}-\frac{9}{x}\right)=64x$$

Since \frac{16x}{x} and \frac{9}{x} have the same denominator, subtract them by subtracting their numerators.

$$9\times \frac{16x-9}{x}=64x$$

Express 9\times \frac{16x-9}{x} as a single fraction.

$$\frac{9\left(16x-9\right)}{x}=64x$$

Use the distributive property to multiply 9 by 16x-9.

$$\frac{144x-81}{x}=64x$$

Subtract 64x from both sides.

$$\frac{144x-81}{x}-64x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -64x times \frac{x}{x}.

$$\frac{144x-81}{x}+\frac{-64xx}{x}=0$$

Since \frac{144x-81}{x} and \frac{-64xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{144x-81-64xx}{x}=0$$

Do the multiplications in 144x-81-64xx.

$$\frac{144x-81-64x^{2}}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$144x-81-64x^{2}=0$$

Add 81 to both sides. Anything plus zero gives itself.

$$144x-64x^{2}=81$$

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.

$$-64x^{2}+144x=81$$

Divide both sides by -64.

$$\frac{-64x^{2}+144x}{-64}=\frac{81}{-64}$$

Dividing by -64 undoes the multiplication by -64.

$$x^{2}+\frac{144}{-64}x=\frac{81}{-64}$$

Reduce the fraction \frac{144}{-64} to lowest terms by extracting and canceling out 16.

$$x^{2}-\frac{9}{4}x=\frac{81}{-64}$$

Divide 81 by -64.

$$x^{2}-\frac{9}{4}x=-\frac{81}{64}$$

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=-\frac{81}{64}+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{-81+81}{64}$$

Add -\frac{81}{64} to \frac{81}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=0$$

Factor x^{2}-\frac{9}{4}x+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{8}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{9}{8}=0 x-\frac{9}{8}=0$$

Add \frac{9}{8} to both sides of the equation.

$$x=\frac{9}{8} x=\frac{9}{8}$$

The equation is now solved. Solutions are the same.

$$x=\frac{9}{8}$$