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

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

Solve for x

$$x=\frac{\sqrt{29}+5}{18}\approx 0.5769536$$

$$x=\frac{5-\sqrt{29}}{18}\approx -0.021398045$$

Solution Steps

Square both sides of the equation.

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

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

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

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

$$81x^{2}+36x+4=81x+5$$

Subtract 81x from both sides.

$$81x^{2}+36x+4-81x=5$$

Combine 36x and -81x to get -45x.

$$81x^{2}-45x+4=5$$

Subtract 5 from both sides.

$$81x^{2}-45x+4-5=0$$

Subtract 5 from 4 to get -1.

$$81x^{2}-45x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, -45 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\times 81\left(-1\right)}}{2\times 81}$$

Square -45.

$$x=\frac{-\left(-45\right)±\sqrt{2025-4\times 81\left(-1\right)}}{2\times 81}$$

Multiply -4 times 81.

$$x=\frac{-\left(-45\right)±\sqrt{2025-324\left(-1\right)}}{2\times 81}$$

Multiply -324 times -1.

$$x=\frac{-\left(-45\right)±\sqrt{2025+324}}{2\times 81}$$

Add 2025 to 324.

$$x=\frac{-\left(-45\right)±\sqrt{2349}}{2\times 81}$$

Take the square root of 2349.

$$x=\frac{-\left(-45\right)±9\sqrt{29}}{2\times 81}$$

The opposite of -45 is 45.

$$x=\frac{45±9\sqrt{29}}{2\times 81}$$

Multiply 2 times 81.

$$x=\frac{45±9\sqrt{29}}{162}$$

Now solve the equation x=\frac{45±9\sqrt{29}}{162} when ± is plus. Add 45 to 9\sqrt{29}.

$$x=\frac{9\sqrt{29}+45}{162}$$

Divide 45+9\sqrt{29} by 162.

$$x=\frac{\sqrt{29}+5}{18}$$

Now solve the equation x=\frac{45±9\sqrt{29}}{162} when ± is minus. Subtract 9\sqrt{29} from 45.

$$x=\frac{45-9\sqrt{29}}{162}$$

Divide 45-9\sqrt{29} by 162.

$$x=\frac{5-\sqrt{29}}{18}$$

The equation is now solved.

$$x=\frac{\sqrt{29}+5}{18} x=\frac{5-\sqrt{29}}{18}$$

Substitute \frac{\sqrt{29}+5}{18} for x in the equation 9x+2=\sqrt{81x+5}.

$$9\times \frac{\sqrt{29}+5}{18}+2=\sqrt{81\times \frac{\sqrt{29}+5}{18}+5}$$

Simplify. The value x=\frac{\sqrt{29}+5}{18} satisfies the equation.

$$\frac{1}{2}\times 29^{\frac{1}{2}}+\frac{9}{2}=\frac{9}{2}+\frac{1}{2}\times 29^{\frac{1}{2}}$$

Substitute \frac{5-\sqrt{29}}{18} for x in the equation 9x+2=\sqrt{81x+5}.

$$9\times \frac{5-\sqrt{29}}{18}+2=\sqrt{81\times \frac{5-\sqrt{29}}{18}+5}$$

Simplify. The value x=\frac{5-\sqrt{29}}{18} satisfies the equation.

$$\frac{9}{2}-\frac{1}{2}\times 29^{\frac{1}{2}}=\frac{9}{2}-\frac{1}{2}\times 29^{\frac{1}{2}}$$

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

$$x=\frac{\sqrt{29}+5}{18} x=\frac{5-\sqrt{29}}{18}$$