Solve 9x+2=3sqrt{81x+5}

$$9x+2 = 3 \sqrt{ 81x+5 }$$

Solve for x

$$x=\frac{\sqrt{677}}{6}+\frac{77}{18}\approx 8.614315055$$

$$x=-\frac{\sqrt{677}}{6}+\frac{77}{18}\approx -0.058759499$$

Solution Steps

Square both sides of the equation.

$$\left(9x+2\right)^{2}=\left(3\sqrt{81x+5}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9x+2\right)^{2}.

$$81x^{2}+36x+4=\left(3\sqrt{81x+5}\right)^{2}$$

Expand \left(3\sqrt{81x+5}\right)^{2}.

$$81x^{2}+36x+4=3^{2}\left(\sqrt{81x+5}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$81x^{2}+36x+4=9\left(\sqrt{81x+5}\right)^{2}$$

Calculate \sqrt{81x+5} to the power of 2 and get 81x+5.

$$81x^{2}+36x+4=9\left(81x+5\right)$$

Use the distributive property to multiply 9 by 81x+5.

$$81x^{2}+36x+4=729x+45$$

Subtract 729x from both sides.

$$81x^{2}+36x+4-729x=45$$

Combine 36x and -729x to get -693x.

$$81x^{2}-693x+4=45$$

Subtract 45 from both sides.

$$81x^{2}-693x+4-45=0$$

Subtract 45 from 4 to get -41.

$$81x^{2}-693x-41=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, -693 for b, and -41 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-693\right)±\sqrt{\left(-693\right)^{2}-4\times 81\left(-41\right)}}{2\times 81}$$

Square -693.

$$x=\frac{-\left(-693\right)±\sqrt{480249-4\times 81\left(-41\right)}}{2\times 81}$$

Multiply -4 times 81.

$$x=\frac{-\left(-693\right)±\sqrt{480249-324\left(-41\right)}}{2\times 81}$$

Multiply -324 times -41.

$$x=\frac{-\left(-693\right)±\sqrt{480249+13284}}{2\times 81}$$

Add 480249 to 13284.

$$x=\frac{-\left(-693\right)±\sqrt{493533}}{2\times 81}$$

Take the square root of 493533.

$$x=\frac{-\left(-693\right)±27\sqrt{677}}{2\times 81}$$

The opposite of -693 is 693.

$$x=\frac{693±27\sqrt{677}}{2\times 81}$$

Multiply 2 times 81.

$$x=\frac{693±27\sqrt{677}}{162}$$

Now solve the equation x=\frac{693±27\sqrt{677}}{162} when ± is plus. Add 693 to 27\sqrt{677}.

$$x=\frac{27\sqrt{677}+693}{162}$$

Divide 693+27\sqrt{677} by 162.

$$x=\frac{\sqrt{677}}{6}+\frac{77}{18}$$

Now solve the equation x=\frac{693±27\sqrt{677}}{162} when ± is minus. Subtract 27\sqrt{677} from 693.

$$x=\frac{693-27\sqrt{677}}{162}$$

Divide 693-27\sqrt{677} by 162.

$$x=-\frac{\sqrt{677}}{6}+\frac{77}{18}$$

The equation is now solved.

$$x=\frac{\sqrt{677}}{6}+\frac{77}{18} x=-\frac{\sqrt{677}}{6}+\frac{77}{18}$$

Substitute \frac{\sqrt{677}}{6}+\frac{77}{18} for x in the equation 9x+2=3\sqrt{81x+5}.

$$9\left(\frac{\sqrt{677}}{6}+\frac{77}{18}\right)+2=3\sqrt{81\left(\frac{\sqrt{677}}{6}+\frac{77}{18}\right)+5}$$

Simplify. The value x=\frac{\sqrt{677}}{6}+\frac{77}{18} satisfies the equation.

$$\frac{3}{2}\times 677^{\frac{1}{2}}+\frac{81}{2}=\frac{81}{2}+\frac{3}{2}\times 677^{\frac{1}{2}}$$

Substitute -\frac{\sqrt{677}}{6}+\frac{77}{18} for x in the equation 9x+2=3\sqrt{81x+5}.

$$9\left(-\frac{\sqrt{677}}{6}+\frac{77}{18}\right)+2=3\sqrt{81\left(-\frac{\sqrt{677}}{6}+\frac{77}{18}\right)+5}$$

Simplify. The value x=-\frac{\sqrt{677}}{6}+\frac{77}{18} satisfies the equation.

$$-\frac{3}{2}\times 677^{\frac{1}{2}}+\frac{81}{2}=\frac{81}{2}-\frac{3}{2}\times 677^{\frac{1}{2}}$$

List all solutions of 9x+2=3\sqrt{81x+5}.

$$x=\frac{\sqrt{677}}{6}+\frac{77}{18} x=-\frac{\sqrt{677}}{6}+\frac{77}{18}$$