Solve sqrt{a/9}+5/2sqrt{4a}=6

$$\sqrt { \frac { a } { 9 } } + \frac { 5 } { 2 } \sqrt { 4 a } = 6$$

Solve for a

$$a = \frac{81}{64} = 1\frac{17}{64} = 1.265625$$

Solution Steps

Subtract \frac{5}{2}\sqrt{4a} from both sides of the equation.

$$\sqrt{\frac{a}{9}}=6-\frac{5}{2}\sqrt{4a}$$

Square both sides of the equation.

$$\left(\sqrt{\frac{a}{9}}\right)^{2}=\left(6-\frac{5}{2}\sqrt{4a}\right)^{2}$$

Calculate \sqrt{\frac{a}{9}} to the power of 2 and get \frac{a}{9}.

$$\frac{a}{9}=\left(6-\frac{5}{2}\sqrt{4a}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-\frac{5}{2}\sqrt{4a}\right)^{2}.

$$\frac{a}{9}=36-30\sqrt{4a}+\frac{25}{4}\left(\sqrt{4a}\right)^{2}$$

Calculate \sqrt{4a} to the power of 2 and get 4a.

$$\frac{a}{9}=36-30\sqrt{4a}+\frac{25}{4}\times 4a$$

Cancel out 4 and 4.

$$\frac{a}{9}=36-30\sqrt{4a}+25a$$

Multiply both sides of the equation by 9.

$$a=324-270\sqrt{4a}+225a$$

Subtract 324+225a from both sides of the equation.

$$a-\left(324+225a\right)=-270\sqrt{4a}$$

To find the opposite of 324+225a, find the opposite of each term.

$$a-324-225a=-270\sqrt{4a}$$

Combine a and -225a to get -224a.

$$-224a-324=-270\sqrt{4a}$$

Square both sides of the equation.

$$\left(-224a-324\right)^{2}=\left(-270\sqrt{4a}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-224a-324\right)^{2}.

$$50176a^{2}+145152a+104976=\left(-270\sqrt{4a}\right)^{2}$$

Expand \left(-270\sqrt{4a}\right)^{2}.

$$50176a^{2}+145152a+104976=\left(-270\right)^{2}\left(\sqrt{4a}\right)^{2}$$

Calculate -270 to the power of 2 and get 72900.

$$50176a^{2}+145152a+104976=72900\left(\sqrt{4a}\right)^{2}$$

Calculate \sqrt{4a} to the power of 2 and get 4a.

$$50176a^{2}+145152a+104976=72900\times 4a$$

Multiply 72900 and 4 to get 291600.

$$50176a^{2}+145152a+104976=291600a$$

Subtract 291600a from both sides.

$$50176a^{2}+145152a+104976-291600a=0$$

Combine 145152a and -291600a to get -146448a.

$$50176a^{2}-146448a+104976=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 50176 for a, -146448 for b, and 104976 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-146448\right)±\sqrt{\left(-146448\right)^{2}-4\times 50176\times 104976}}{2\times 50176}$$

Square -146448.

$$a=\frac{-\left(-146448\right)±\sqrt{21447016704-4\times 50176\times 104976}}{2\times 50176}$$

Multiply -4 times 50176.

$$a=\frac{-\left(-146448\right)±\sqrt{21447016704-200704\times 104976}}{2\times 50176}$$

Multiply -200704 times 104976.

$$a=\frac{-\left(-146448\right)±\sqrt{21447016704-21069103104}}{2\times 50176}$$

Add 21447016704 to -21069103104.

$$a=\frac{-\left(-146448\right)±\sqrt{377913600}}{2\times 50176}$$

Take the square root of 377913600.

$$a=\frac{-\left(-146448\right)±19440}{2\times 50176}$$

The opposite of -146448 is 146448.

$$a=\frac{146448±19440}{2\times 50176}$$

Multiply 2 times 50176.

$$a=\frac{146448±19440}{100352}$$

Now solve the equation a=\frac{146448±19440}{100352} when ± is plus. Add 146448 to 19440.

$$a=\frac{165888}{100352}$$

Reduce the fraction \frac{165888}{100352} to lowest terms by extracting and canceling out 2048.

$$a=\frac{81}{49}$$

Now solve the equation a=\frac{146448±19440}{100352} when ± is minus. Subtract 19440 from 146448.

$$a=\frac{127008}{100352}$$

Reduce the fraction \frac{127008}{100352} to lowest terms by extracting and canceling out 1568.

$$a=\frac{81}{64}$$

The equation is now solved.

$$a=\frac{81}{49} a=\frac{81}{64}$$

Substitute \frac{81}{49} for a in the equation \sqrt{\frac{a}{9}}+\frac{5}{2}\sqrt{4a}=6.

$$\sqrt{\frac{\frac{81}{49}}{9}}+\frac{5}{2}\sqrt{4\times \frac{81}{49}}=6$$

Simplify. The value a=\frac{81}{49} does not satisfy the equation.

$$\frac{48}{7}=6$$

Substitute \frac{81}{64} for a in the equation \sqrt{\frac{a}{9}}+\frac{5}{2}\sqrt{4a}=6.

$$\sqrt{\frac{\frac{81}{64}}{9}}+\frac{5}{2}\sqrt{4\times \frac{81}{64}}=6$$

Simplify. The value a=\frac{81}{64} satisfies the equation.

$$6=6$$

Equation \sqrt{\frac{a}{9}}=-\frac{5\sqrt{4a}}{2}+6 has a unique solution.

$$a=\frac{81}{64}$$