Solve sqrt{9x^2-2x}=3x+4

$$\sqrt{ 9 { x }^{ 2 } -2x } =3x+4$$

Solve for x

$$x=-\frac{8}{13}\approx -0.615384615$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{9x^{2}-2x}\right)^{2}=\left(3x+4\right)^{2}$$

Calculate \sqrt{9x^{2}-2x} to the power of 2 and get 9x^{2}-2x.

$$9x^{2}-2x=\left(3x+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+4\right)^{2}.

$$9x^{2}-2x=9x^{2}+24x+16$$

Subtract 9x^{2} from both sides.

$$9x^{2}-2x-9x^{2}=24x+16$$

Combine 9x^{2} and -9x^{2} to get 0.

$$-2x=24x+16$$

Subtract 24x from both sides.

$$-2x-24x=16$$

Combine -2x and -24x to get -26x.

$$-26x=16$$

Divide both sides by -26.

$$x=\frac{16}{-26}$$

Reduce the fraction \frac{16}{-26} to lowest terms by extracting and canceling out 2.

$$x=-\frac{8}{13}$$

Substitute -\frac{8}{13} for x in the equation \sqrt{9x^{2}-2x}=3x+4.

$$\sqrt{9\left(-\frac{8}{13}\right)^{2}-2\left(-\frac{8}{13}\right)}=3\left(-\frac{8}{13}\right)+4$$

Simplify. The value x=-\frac{8}{13} satisfies the equation.

$$\frac{28}{13}=\frac{28}{13}$$

Equation \sqrt{9x^{2}-2x}=3x+4 has a unique solution.

$$x=-\frac{8}{13}$$