Solve (2a+3/2b)^2+7/4b^2+6a/b=4a^2+1/2b^2

$$( 2 a + \frac { 3 } { 2 b } ) ^ { 2 } + \frac { 7 } { 4 b ^ { 2 } } + \frac { 6 a } { b } = 4 a ^ { 2 } + \frac { 1 } { 2 b ^ { 2 } }$$

Solve for a

$$a=-\frac{7}{24b}$$

$$b\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4b^{2}, the least common multiple of 4b^{2},b,2b^{2}.

$$4b^{2}\left(2a+\frac{3}{2b}\right)^{2}+7+4b\times 6a=4a^{2}\times 4b^{2}+2$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2a times \frac{2b}{2b}.

$$4b^{2}\left(\frac{2a\times 2b}{2b}+\frac{3}{2b}\right)^{2}+7+4b\times 6a=4a^{2}\times 4b^{2}+2$$

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

$$4b^{2}\times \left(\frac{2a\times 2b+3}{2b}\right)^{2}+7+4b\times 6a=4a^{2}\times 4b^{2}+2$$

Do the multiplications in 2a\times 2b+3.

$$4b^{2}\times \left(\frac{4ab+3}{2b}\right)^{2}+7+4b\times 6a=4a^{2}\times 4b^{2}+2$$

To raise \frac{4ab+3}{2b} to a power, raise both numerator and denominator to the power and then divide.

$$4b^{2}\times \frac{\left(4ab+3\right)^{2}}{\left(2b\right)^{2}}+7+4b\times 6a=4a^{2}\times 4b^{2}+2$$

Express 4\times \frac{\left(4ab+3\right)^{2}}{\left(2b\right)^{2}} as a single fraction.

$$\frac{4\left(4ab+3\right)^{2}}{\left(2b\right)^{2}}b^{2}+7+4b\times 6a=4a^{2}\times 4b^{2}+2$$

Multiply 4 and 6 to get 24.

$$\frac{4\left(4ab+3\right)^{2}}{\left(2b\right)^{2}}b^{2}+7+24ba=4a^{2}\times 4b^{2}+2$$

Multiply 4 and 4 to get 16.

$$\frac{4\left(4ab+3\right)^{2}}{\left(2b\right)^{2}}b^{2}+7+24ba=16a^{2}b^{2}+2$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(4ab+3\right)^{2}.

$$\frac{4\left(16a^{2}b^{2}+24ab+9\right)}{\left(2b\right)^{2}}b^{2}+7+24ba=16a^{2}b^{2}+2$$

Expand \left(2b\right)^{2}.

$$\frac{4\left(16a^{2}b^{2}+24ab+9\right)}{2^{2}b^{2}}b^{2}+7+24ba=16a^{2}b^{2}+2$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4\left(16a^{2}b^{2}+24ab+9\right)}{4b^{2}}b^{2}+7+24ba=16a^{2}b^{2}+2$$

Cancel out 4 in both numerator and denominator.

$$\frac{16a^{2}b^{2}+24ab+9}{b^{2}}b^{2}+7+24ba=16a^{2}b^{2}+2$$

Express \frac{16a^{2}b^{2}+24ab+9}{b^{2}}b^{2} as a single fraction.

$$\frac{\left(16a^{2}b^{2}+24ab+9\right)b^{2}}{b^{2}}+7+24ba=16a^{2}b^{2}+2$$

Cancel out b^{2} in both numerator and denominator.

$$16a^{2}b^{2}+24ab+9+7+24ba=16a^{2}b^{2}+2$$

Add 9 and 7 to get 16.

$$16a^{2}b^{2}+24ab+16+24ba=16a^{2}b^{2}+2$$

Combine 24ab and 24ba to get 48ab.

$$16a^{2}b^{2}+48ab+16=16a^{2}b^{2}+2$$

Subtract 16a^{2}b^{2} from both sides.

$$16a^{2}b^{2}+48ab+16-16a^{2}b^{2}=2$$

Combine 16a^{2}b^{2} and -16a^{2}b^{2} to get 0.

$$48ab+16=2$$

Subtract 16 from both sides.

$$48ab=2-16$$

Subtract 16 from 2 to get -14.

$$48ab=-14$$

The equation is in standard form.

$$48ba=-14$$

Divide both sides by 48b.

$$\frac{48ba}{48b}=-\frac{14}{48b}$$

Dividing by 48b undoes the multiplication by 48b.

$$a=-\frac{14}{48b}$$

Divide -14 by 48b.

$$a=-\frac{7}{24b}$$

Solve for b

$$b=-\frac{7}{24a}$$

$$a\neq 0$$