Solve 9(x+1)=sqrt{2x+5}

$$9 ( x + 1 ) = \sqrt { 2 x + 5 }$$

Solve for x

$$x=\frac{2\left(\sqrt{61}-40\right)}{81}\approx -0.79480865$$

Solution Steps

Square both sides of the equation.

$$\left(9\left(x+1\right)\right)^{2}=\left(\sqrt{2x+5}\right)^{2}$$

Use the distributive property to multiply 9 by x+1.

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

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

$$81x^{2}+162x+81=\left(\sqrt{2x+5}\right)^{2}$$

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

$$81x^{2}+162x+81=2x+5$$

Subtract 2x from both sides.

$$81x^{2}+162x+81-2x=5$$

Combine 162x and -2x to get 160x.

$$81x^{2}+160x+81=5$$

Subtract 5 from both sides.

$$81x^{2}+160x+81-5=0$$

Subtract 5 from 81 to get 76.

$$81x^{2}+160x+76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, 160 for b, and 76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-160±\sqrt{160^{2}-4\times 81\times 76}}{2\times 81}$$

Square 160.

$$x=\frac{-160±\sqrt{25600-4\times 81\times 76}}{2\times 81}$$

Multiply -4 times 81.

$$x=\frac{-160±\sqrt{25600-324\times 76}}{2\times 81}$$

Multiply -324 times 76.

$$x=\frac{-160±\sqrt{25600-24624}}{2\times 81}$$

Add 25600 to -24624.

$$x=\frac{-160±\sqrt{976}}{2\times 81}$$

Take the square root of 976.

$$x=\frac{-160±4\sqrt{61}}{2\times 81}$$

Multiply 2 times 81.

$$x=\frac{-160±4\sqrt{61}}{162}$$

Now solve the equation x=\frac{-160±4\sqrt{61}}{162} when ± is plus. Add -160 to 4\sqrt{61}.

$$x=\frac{4\sqrt{61}-160}{162}$$

Divide -160+4\sqrt{61} by 162.

$$x=\frac{2\sqrt{61}-80}{81}$$

Now solve the equation x=\frac{-160±4\sqrt{61}}{162} when ± is minus. Subtract 4\sqrt{61} from -160.

$$x=\frac{-4\sqrt{61}-160}{162}$$

Divide -160-4\sqrt{61} by 162.

$$x=\frac{-2\sqrt{61}-80}{81}$$

The equation is now solved.

$$x=\frac{2\sqrt{61}-80}{81} x=\frac{-2\sqrt{61}-80}{81}$$

Substitute \frac{2\sqrt{61}-80}{81} for x in the equation 9\left(x+1\right)=\sqrt{2x+5}.

$$9\left(\frac{2\sqrt{61}-80}{81}+1\right)=\sqrt{2\times \frac{2\sqrt{61}-80}{81}+5}$$

Simplify. The value x=\frac{2\sqrt{61}-80}{81} satisfies the equation.

$$\frac{2}{9}\times 61^{\frac{1}{2}}+\frac{1}{9}=\frac{2}{9}\times 61^{\frac{1}{2}}+\frac{1}{9}$$

Substitute \frac{-2\sqrt{61}-80}{81} for x in the equation 9\left(x+1\right)=\sqrt{2x+5}.

$$9\left(\frac{-2\sqrt{61}-80}{81}+1\right)=\sqrt{2\times \frac{-2\sqrt{61}-80}{81}+5}$$

Simplify. The value x=\frac{-2\sqrt{61}-80}{81} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{2}{9}\times 61^{\frac{1}{2}}+\frac{1}{9}=\frac{2}{9}\times 61^{\frac{1}{2}}-\frac{1}{9}$$

Equation 9\left(x+1\right)=\sqrt{2x+5} has a unique solution.

$$x=\frac{2\sqrt{61}-80}{81}$$