Solve 8(2/3+3a^2/2)/24=44

$$\frac { 8 ( \frac { 2 } { 3 } + \frac { 3 a ^ { 2 } } { 2 } ) } { 24 } = 44$$

Solve for a

$$a = \frac{2 \sqrt{197}}{3} \approx 9.357112565$$

$$a = -\frac{2 \sqrt{197}}{3} \approx -9.357112565$$

Steps by Finding Square Root

Multiply both sides by 24.

$$8\left(\frac{2}{3}+\frac{3a^{2}}{2}\right)=44\times 24$$

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$48\left(\frac{2}{3}+\frac{3a^{2}}{2}\right)=264\times 24$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2}{3} times \frac{2}{2}. Multiply \frac{3a^{2}}{2} times \frac{3}{3}.

$$48\left(\frac{2\times 2}{6}+\frac{3\times 3a^{2}}{6}\right)=264\times 24$$

Since \frac{2\times 2}{6} and \frac{3\times 3a^{2}}{6} have the same denominator, add them by adding their numerators.

$$48\times \frac{2\times 2+3\times 3a^{2}}{6}=264\times 24$$

Do the multiplications in 2\times 2+3\times 3a^{2}.

$$48\times \frac{4+9a^{2}}{6}=264\times 24$$

Cancel out 6, the greatest common factor in 48 and 6.

$$8\left(4+9a^{2}\right)=264\times 24$$

Use the distributive property to multiply 8 by 4+9a^{2}.

$$32+72a^{2}=264\times 24$$

Multiply 264 and 24 to get 6336.

$$32+72a^{2}=6336$$

Subtract 32 from both sides.

$$72a^{2}=6336-32$$

Subtract 32 from 6336 to get 6304.

$$72a^{2}=6304$$

Divide both sides by 72.

$$a^{2}=\frac{6304}{72}$$

Reduce the fraction \frac{6304}{72} to lowest terms by extracting and canceling out 8.

$$a^{2}=\frac{788}{9}$$

Take the square root of both sides of the equation.

$$a=\frac{2\sqrt{197}}{3} a=-\frac{2\sqrt{197}}{3}$$

Steps Using the Quadratic Formula

Multiply both sides by 24.

$$8\left(\frac{2}{3}+\frac{3a^{2}}{2}\right)=44\times 24$$

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$48\left(\frac{2}{3}+\frac{3a^{2}}{2}\right)=264\times 24$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2}{3} times \frac{2}{2}. Multiply \frac{3a^{2}}{2} times \frac{3}{3}.

$$48\left(\frac{2\times 2}{6}+\frac{3\times 3a^{2}}{6}\right)=264\times 24$$

Since \frac{2\times 2}{6} and \frac{3\times 3a^{2}}{6} have the same denominator, add them by adding their numerators.

$$48\times \frac{2\times 2+3\times 3a^{2}}{6}=264\times 24$$

Do the multiplications in 2\times 2+3\times 3a^{2}.

$$48\times \frac{4+9a^{2}}{6}=264\times 24$$

Cancel out 6, the greatest common factor in 48 and 6.

$$8\left(4+9a^{2}\right)=264\times 24$$

Use the distributive property to multiply 8 by 4+9a^{2}.

$$32+72a^{2}=264\times 24$$

Multiply 264 and 24 to get 6336.

$$32+72a^{2}=6336$$

Subtract 6336 from both sides.

$$32+72a^{2}-6336=0$$

Subtract 6336 from 32 to get -6304.

$$-6304+72a^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$72a^{2}-6304=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 72 for a, 0 for b, and -6304 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{0±\sqrt{0^{2}-4\times 72\left(-6304\right)}}{2\times 72}$$

Square 0.

$$a=\frac{0±\sqrt{-4\times 72\left(-6304\right)}}{2\times 72}$$

Multiply -4 times 72.

$$a=\frac{0±\sqrt{-288\left(-6304\right)}}{2\times 72}$$

Multiply -288 times -6304.

$$a=\frac{0±\sqrt{1815552}}{2\times 72}$$

Take the square root of 1815552.

$$a=\frac{0±96\sqrt{197}}{2\times 72}$$

Multiply 2 times 72.

$$a=\frac{0±96\sqrt{197}}{144}$$

Now solve the equation a=\frac{0±96\sqrt{197}}{144} when ± is plus.

$$a=\frac{2\sqrt{197}}{3}$$

Now solve the equation a=\frac{0±96\sqrt{197}}{144} when ± is minus.

$$a=-\frac{2\sqrt{197}}{3}$$

The equation is now solved.

$$a=\frac{2\sqrt{197}}{3} a=-\frac{2\sqrt{197}}{3}$$