Solve 8/9+9/8=x+2sqrt{x}

$$\frac{ 8 }{ 9 } + \frac{ 9 }{ 8 } =x+2 \sqrt{ x }$$

Solve for x

$$x=-\frac{\sqrt{434}}{6}+\frac{289}{72}\approx 0.54177778$$

Solution Steps

Subtract x from both sides of the equation.

$$\frac{8}{9}+\frac{9}{8}-x=2\sqrt{x}$$

Least common multiple of 9 and 8 is 72. Convert \frac{8}{9} and \frac{9}{8} to fractions with denominator 72.

$$\frac{64}{72}+\frac{81}{72}-x=2\sqrt{x}$$

Since \frac{64}{72} and \frac{81}{72} have the same denominator, add them by adding their numerators.

$$\frac{64+81}{72}-x=2\sqrt{x}$$

Add 64 and 81 to get 145.

$$\frac{145}{72}-x=2\sqrt{x}$$

Square both sides of the equation.

$$\left(\frac{145}{72}-x\right)^{2}=\left(2\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{145}{72}-x\right)^{2}.

$$\frac{21025}{5184}-\frac{145}{36}x+x^{2}=\left(2\sqrt{x}\right)^{2}$$

Expand \left(2\sqrt{x}\right)^{2}.

$$\frac{21025}{5184}-\frac{145}{36}x+x^{2}=2^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{21025}{5184}-\frac{145}{36}x+x^{2}=4\left(\sqrt{x}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{21025}{5184}-\frac{145}{36}x+x^{2}=4x$$

Subtract 4x from both sides.

$$\frac{21025}{5184}-\frac{145}{36}x+x^{2}-4x=0$$

Combine -\frac{145}{36}x and -4x to get -\frac{289}{36}x.

$$\frac{21025}{5184}-\frac{289}{36}x+x^{2}=0$$

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.

$$x^{2}-\frac{289}{36}x+\frac{21025}{5184}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{289}{36} for b, and \frac{21025}{5184} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{289}{36}\right)±\sqrt{\left(-\frac{289}{36}\right)^{2}-4\times \frac{21025}{5184}}}{2}$$

Square -\frac{289}{36} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{289}{36}\right)±\sqrt{\frac{83521}{1296}-4\times \frac{21025}{5184}}}{2}$$

Multiply -4 times \frac{21025}{5184}.

$$x=\frac{-\left(-\frac{289}{36}\right)±\sqrt{\frac{83521-21025}{1296}}}{2}$$

Add \frac{83521}{1296} to -\frac{21025}{1296} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{289}{36}\right)±\sqrt{\frac{434}{9}}}{2}$$

Take the square root of \frac{434}{9}.

$$x=\frac{-\left(-\frac{289}{36}\right)±\frac{\sqrt{434}}{3}}{2}$$

The opposite of -\frac{289}{36} is \frac{289}{36}.

$$x=\frac{\frac{289}{36}±\frac{\sqrt{434}}{3}}{2}$$

Now solve the equation x=\frac{\frac{289}{36}±\frac{\sqrt{434}}{3}}{2} when ± is plus. Add \frac{289}{36} to \frac{\sqrt{434}}{3}.

$$x=\frac{\frac{\sqrt{434}}{3}+\frac{289}{36}}{2}$$

Divide \frac{289}{36}+\frac{\sqrt{434}}{3} by 2.

$$x=\frac{\sqrt{434}}{6}+\frac{289}{72}$$

Now solve the equation x=\frac{\frac{289}{36}±\frac{\sqrt{434}}{3}}{2} when ± is minus. Subtract \frac{\sqrt{434}}{3} from \frac{289}{36}.

$$x=\frac{-\frac{\sqrt{434}}{3}+\frac{289}{36}}{2}$$

Divide \frac{289}{36}-\frac{\sqrt{434}}{3} by 2.

$$x=-\frac{\sqrt{434}}{6}+\frac{289}{72}$$

The equation is now solved.

$$x=\frac{\sqrt{434}}{6}+\frac{289}{72} x=-\frac{\sqrt{434}}{6}+\frac{289}{72}$$

Substitute \frac{\sqrt{434}}{6}+\frac{289}{72} for x in the equation \frac{8}{9}+\frac{9}{8}=x+2\sqrt{x}.

$$\frac{8}{9}+\frac{9}{8}=\frac{\sqrt{434}}{6}+\frac{289}{72}+2\sqrt{\frac{\sqrt{434}}{6}+\frac{289}{72}}$$

Simplify. The value x=\frac{\sqrt{434}}{6}+\frac{289}{72} does not satisfy the equation.

$$\frac{145}{72}=\frac{1}{3}\times 434^{\frac{1}{2}}+\frac{433}{72}$$

Substitute -\frac{\sqrt{434}}{6}+\frac{289}{72} for x in the equation \frac{8}{9}+\frac{9}{8}=x+2\sqrt{x}.

$$\frac{8}{9}+\frac{9}{8}=-\frac{\sqrt{434}}{6}+\frac{289}{72}+2\sqrt{-\frac{\sqrt{434}}{6}+\frac{289}{72}}$$

Simplify. The value x=-\frac{\sqrt{434}}{6}+\frac{289}{72} satisfies the equation.

$$\frac{145}{72}=\frac{145}{72}$$

Equation \frac{145}{72}-x=2\sqrt{x} has a unique solution.

$$x=-\frac{\sqrt{434}}{6}+\frac{289}{72}$$