Solve 2*(2x+3)+sqrt{a}(x-3/2)-7/2=0

$$2 \times ( 2 x + 3 ) + \sqrt { a } ( x - \frac { 3 } { 2 } ) - \frac { 7 } { 2 } = 0$$

Solve for a

$$a=\left(\frac{8x+5}{2x-3}\right)^{2}$$

$$x\geq -\frac{5}{8}\text{ and }x<\frac{3}{2}$$

Steps for Solving Radical Equations

Subtract 4x+\frac{5}{2} from both sides of the equation.

$$\left(x-\frac{3}{2}\right)\sqrt{a}+4x+\frac{5}{2}-\left(4x+\frac{5}{2}\right)=-\left(4x+\frac{5}{2}\right)$$

Subtracting 4x+\frac{5}{2} from itself leaves 0.

$$\left(x-\frac{3}{2}\right)\sqrt{a}=-\left(4x+\frac{5}{2}\right)$$

Subtract 4x+\frac{5}{2} from 0.

$$\left(x-\frac{3}{2}\right)\sqrt{a}=-4x-\frac{5}{2}$$

Divide both sides by x-\frac{3}{2}.

$$\frac{\left(x-\frac{3}{2}\right)\sqrt{a}}{x-\frac{3}{2}}=\frac{-4x-\frac{5}{2}}{x-\frac{3}{2}}$$

Dividing by x-\frac{3}{2} undoes the multiplication by x-\frac{3}{2}.

$$\sqrt{a}=\frac{-4x-\frac{5}{2}}{x-\frac{3}{2}}$$

Divide -4x-\frac{5}{2} by x-\frac{3}{2}.

$$\sqrt{a}=-\frac{8x+5}{2x-3}$$

Square both sides of the equation.

$$a=\frac{\left(8x+5\right)^{2}}{\left(2x-3\right)^{2}}$$

Solve for x

$$x=-\frac{-3\sqrt{a}+5}{2\left(\sqrt{a}+4\right)}$$

$$a\geq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2 by 2x+3.

$$4x+6+\sqrt{a}\left(x-\frac{3}{2}\right)-\frac{7}{2}=0$$

Use the distributive property to multiply \sqrt{a} by x-\frac{3}{2}.

$$4x+6+\sqrt{a}x-\frac{3}{2}\sqrt{a}-\frac{7}{2}=0$$

Subtract \frac{7}{2} from 6 to get \frac{5}{2}.

$$4x+\frac{5}{2}+\sqrt{a}x-\frac{3}{2}\sqrt{a}=0$$

Subtract \frac{5}{2} from both sides. Anything subtracted from zero gives its negation.

$$4x+\sqrt{a}x-\frac{3}{2}\sqrt{a}=-\frac{5}{2}$$

Add \frac{3}{2}\sqrt{a} to both sides.

$$4x+\sqrt{a}x=-\frac{5}{2}+\frac{3}{2}\sqrt{a}$$

Combine all terms containing x.

$$\left(4+\sqrt{a}\right)x=-\frac{5}{2}+\frac{3}{2}\sqrt{a}$$

The equation is in standard form.

$$\left(\sqrt{a}+4\right)x=\frac{3\sqrt{a}-5}{2}$$

Divide both sides by 4+\sqrt{a}.

$$\frac{\left(\sqrt{a}+4\right)x}{\sqrt{a}+4}=\frac{3\sqrt{a}-5}{2\left(\sqrt{a}+4\right)}$$

Dividing by 4+\sqrt{a} undoes the multiplication by 4+\sqrt{a}.

$$x=\frac{3\sqrt{a}-5}{2\left(\sqrt{a}+4\right)}$$