Solve sqrt{5x+9}=2x+3

$$\sqrt { 5 x + 9 } = 2 x + 3$$

Solve for x

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5x+9}\right)^{2}=\left(2x+3\right)^{2}$$

Calculate \sqrt{5x+9} to the power of 2 and get 5x+9.

$$5x+9=\left(2x+3\right)^{2}$$

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

$$5x+9=4x^{2}+12x+9$$

Subtract 4x^{2} from both sides.

$$5x+9-4x^{2}=12x+9$$

Subtract 12x from both sides.

$$5x+9-4x^{2}-12x=9$$

Combine 5x and -12x to get -7x.

$$-7x+9-4x^{2}=9$$

Subtract 9 from both sides.

$$-7x+9-4x^{2}-9=0$$

Subtract 9 from 9 to get 0.

$$-7x-4x^{2}=0$$

Factor out x.

$$x\left(-7-4x\right)=0$$

To find equation solutions, solve x=0 and -7-4x=0.

$$x=0 x=-\frac{7}{4}$$

Substitute 0 for x in the equation \sqrt{5x+9}=2x+3.

$$\sqrt{5\times 0+9}=2\times 0+3$$

Simplify. The value x=0 satisfies the equation.

$$3=3$$

Substitute -\frac{7}{4} for x in the equation \sqrt{5x+9}=2x+3.

$$\sqrt{5\left(-\frac{7}{4}\right)+9}=2\left(-\frac{7}{4}\right)+3$$

Simplify. The value x=-\frac{7}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{2}=-\frac{1}{2}$$

Equation \sqrt{5x+9}=2x+3 has a unique solution.

$$x=0$$