Solve 3/2n+-3/5sqrt{n}=1.45

$$\frac{ 3 }{ 2 } n+ \frac{ -3 }{ 5 } \sqrt{ n } = 1.45$$

Solve for n

$$n=\frac{\sqrt{906}}{75}+\frac{157}{150}\approx 1.447997785$$

Solution Steps

Subtract \frac{3}{2}n from both sides of the equation.

$$-\frac{3}{5}\sqrt{n}=1.45-\frac{3}{2}n$$

Square both sides of the equation.

$$\left(-\frac{3}{5}\sqrt{n}\right)^{2}=\left(1.45-\frac{3}{2}n\right)^{2}$$

Expand \left(-\frac{3}{5}\sqrt{n}\right)^{2}.

$$\left(-\frac{3}{5}\right)^{2}\left(\sqrt{n}\right)^{2}=\left(1.45-\frac{3}{2}n\right)^{2}$$

Calculate -\frac{3}{5} to the power of 2 and get \frac{9}{25}.

$$\frac{9}{25}\left(\sqrt{n}\right)^{2}=\left(1.45-\frac{3}{2}n\right)^{2}$$

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

$$\frac{9}{25}n=\left(1.45-\frac{3}{2}n\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1.45-\frac{3}{2}n\right)^{2}.

$$\frac{9}{25}n=2.1025-\frac{87}{20}n+\frac{9}{4}n^{2}$$

Subtract 2.1025 from both sides.

$$\frac{9}{25}n-2.1025=-\frac{87}{20}n+\frac{9}{4}n^{2}$$

Add \frac{87}{20}n to both sides.

$$\frac{9}{25}n-2.1025+\frac{87}{20}n=\frac{9}{4}n^{2}$$

Combine \frac{9}{25}n and \frac{87}{20}n to get \frac{471}{100}n.

$$\frac{471}{100}n-2.1025=\frac{9}{4}n^{2}$$

Subtract \frac{9}{4}n^{2} from both sides.

$$\frac{471}{100}n-2.1025-\frac{9}{4}n^{2}=0$$

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.

$$-\frac{9}{4}n^{2}+\frac{471}{100}n-2.1025=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{9}{4} for a, \frac{471}{100} for b, and -2.1025 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\frac{471}{100}±\sqrt{\left(\frac{471}{100}\right)^{2}-4\left(-\frac{9}{4}\right)\left(-2.1025\right)}}{2\left(-\frac{9}{4}\right)}$$

Square \frac{471}{100} by squaring both the numerator and the denominator of the fraction.

$$n=\frac{-\frac{471}{100}±\sqrt{\frac{221841}{10000}-4\left(-\frac{9}{4}\right)\left(-2.1025\right)}}{2\left(-\frac{9}{4}\right)}$$

Multiply -4 times -\frac{9}{4}.

$$n=\frac{-\frac{471}{100}±\sqrt{\frac{221841}{10000}+9\left(-2.1025\right)}}{2\left(-\frac{9}{4}\right)}$$

Multiply 9 times -2.1025.

$$n=\frac{-\frac{471}{100}±\sqrt{\frac{221841}{10000}-18.9225}}{2\left(-\frac{9}{4}\right)}$$

Add \frac{221841}{10000} to -18.9225 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n=\frac{-\frac{471}{100}±\sqrt{\frac{4077}{1250}}}{2\left(-\frac{9}{4}\right)}$$

Take the square root of \frac{4077}{1250}.

$$n=\frac{-\frac{471}{100}±\frac{3\sqrt{906}}{50}}{2\left(-\frac{9}{4}\right)}$$

Multiply 2 times -\frac{9}{4}.

$$n=\frac{-\frac{471}{100}±\frac{3\sqrt{906}}{50}}{-\frac{9}{2}}$$

Now solve the equation n=\frac{-\frac{471}{100}±\frac{3\sqrt{906}}{50}}{-\frac{9}{2}} when ± is plus. Add -\frac{471}{100} to \frac{3\sqrt{906}}{50}.

$$n=\frac{\frac{3\sqrt{906}}{50}-\frac{471}{100}}{-\frac{9}{2}}$$

Divide -\frac{471}{100}+\frac{3\sqrt{906}}{50} by -\frac{9}{2} by multiplying -\frac{471}{100}+\frac{3\sqrt{906}}{50} by the reciprocal of -\frac{9}{2}.

$$n=-\frac{\sqrt{906}}{75}+\frac{157}{150}$$

Now solve the equation n=\frac{-\frac{471}{100}±\frac{3\sqrt{906}}{50}}{-\frac{9}{2}} when ± is minus. Subtract \frac{3\sqrt{906}}{50} from -\frac{471}{100}.

$$n=\frac{-\frac{3\sqrt{906}}{50}-\frac{471}{100}}{-\frac{9}{2}}$$

Divide -\frac{471}{100}-\frac{3\sqrt{906}}{50} by -\frac{9}{2} by multiplying -\frac{471}{100}-\frac{3\sqrt{906}}{50} by the reciprocal of -\frac{9}{2}.

$$n=\frac{\sqrt{906}}{75}+\frac{157}{150}$$

The equation is now solved.

$$n=-\frac{\sqrt{906}}{75}+\frac{157}{150} n=\frac{\sqrt{906}}{75}+\frac{157}{150}$$

Substitute -\frac{\sqrt{906}}{75}+\frac{157}{150} for n in the equation \frac{3}{2}n+\frac{-3}{5}\sqrt{n}=1.45.

$$\frac{3}{2}\left(-\frac{\sqrt{906}}{75}+\frac{157}{150}\right)+\frac{-3}{5}\sqrt{-\frac{\sqrt{906}}{75}+\frac{157}{150}}=1.45$$

Simplify. The value n=-\frac{\sqrt{906}}{75}+\frac{157}{150} does not satisfy the equation.

$$-\frac{1}{25}\times 906^{\frac{1}{2}}+\frac{169}{100}=1.45$$

Substitute \frac{\sqrt{906}}{75}+\frac{157}{150} for n in the equation \frac{3}{2}n+\frac{-3}{5}\sqrt{n}=1.45.

$$\frac{3}{2}\left(\frac{\sqrt{906}}{75}+\frac{157}{150}\right)+\frac{-3}{5}\sqrt{\frac{\sqrt{906}}{75}+\frac{157}{150}}=1.45$$

Simplify. The value n=\frac{\sqrt{906}}{75}+\frac{157}{150} satisfies the equation.

$$\frac{29}{20}=1.45$$

Equation -\frac{3\sqrt{n}}{5}=-\frac{3n}{2}+1.45 has a unique solution.

$$n=\frac{\sqrt{906}}{75}+\frac{157}{150}$$