Solve 5x-3+4=sqrt{9+2x}=

$$5x-3+4= \sqrt{ 9+2x } =$$

Solve for x

$$x=\frac{6\sqrt{6}-4}{25}\approx 0.427877538$$

Solution Steps

Square both sides of the equation.

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

Add -3 and 4 to get 1.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+1\right)^{2}.

$$25x^{2}+10x+1=\left(\sqrt{9+2x}\right)^{2}$$

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

$$25x^{2}+10x+1=9+2x$$

Subtract 9 from both sides.

$$25x^{2}+10x+1-9=2x$$

Subtract 9 from 1 to get -8.

$$25x^{2}+10x-8=2x$$

Subtract 2x from both sides.

$$25x^{2}+10x-8-2x=0$$

Combine 10x and -2x to get 8x.

$$25x^{2}+8x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, 8 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 25\left(-8\right)}}{2\times 25}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 25\left(-8\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-8±\sqrt{64-100\left(-8\right)}}{2\times 25}$$

Multiply -100 times -8.

$$x=\frac{-8±\sqrt{64+800}}{2\times 25}$$

Add 64 to 800.

$$x=\frac{-8±\sqrt{864}}{2\times 25}$$

Take the square root of 864.

$$x=\frac{-8±12\sqrt{6}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-8±12\sqrt{6}}{50}$$

Now solve the equation x=\frac{-8±12\sqrt{6}}{50} when ± is plus. Add -8 to 12\sqrt{6}.

$$x=\frac{12\sqrt{6}-8}{50}$$

Divide -8+12\sqrt{6} by 50.

$$x=\frac{6\sqrt{6}-4}{25}$$

Now solve the equation x=\frac{-8±12\sqrt{6}}{50} when ± is minus. Subtract 12\sqrt{6} from -8.

$$x=\frac{-12\sqrt{6}-8}{50}$$

Divide -8-12\sqrt{6} by 50.

$$x=\frac{-6\sqrt{6}-4}{25}$$

The equation is now solved.

$$x=\frac{6\sqrt{6}-4}{25} x=\frac{-6\sqrt{6}-4}{25}$$

Substitute \frac{6\sqrt{6}-4}{25} for x in the equation 5x-3+4=\sqrt{9+2x}.

$$5\times \frac{6\sqrt{6}-4}{25}-3+4=\sqrt{9+2\times \frac{6\sqrt{6}-4}{25}}$$

Simplify. The value x=\frac{6\sqrt{6}-4}{25} satisfies the equation.

$$\frac{6}{5}\times 6^{\frac{1}{2}}+\frac{1}{5}=\frac{6}{5}\times 6^{\frac{1}{2}}+\frac{1}{5}$$

Substitute \frac{-6\sqrt{6}-4}{25} for x in the equation 5x-3+4=\sqrt{9+2x}.

$$5\times \frac{-6\sqrt{6}-4}{25}-3+4=\sqrt{9+2\times \frac{-6\sqrt{6}-4}{25}}$$

Simplify. The value x=\frac{-6\sqrt{6}-4}{25} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{6}{5}\times 6^{\frac{1}{2}}+\frac{1}{5}=\frac{6}{5}\times 6^{\frac{1}{2}}-\frac{1}{5}$$

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

$$x=\frac{6\sqrt{6}-4}{25}$$