Solve -5/3*x+2=5-sqrt{x+22}-3

$$- \frac { 5 } { 3 } \cdot x + 2 = 5 - \sqrt { x + 22 } - 3$$

Solve for x

$$x=3$$

Solution Steps

Subtract 5-3 from both sides of the equation.

$$-\frac{5}{3}x+2-\left(5-3\right)=-\sqrt{x+22}$$

Subtract 3 from 5 to get 2.

$$-\frac{5}{3}x+2-2=-\sqrt{x+22}$$

Subtract 2 from 2 to get 0.

$$-\frac{5}{3}x=-\sqrt{x+22}$$

Square both sides of the equation.

$$\left(-\frac{5}{3}x\right)^{2}=\left(-\sqrt{x+22}\right)^{2}$$

Expand \left(-\frac{5}{3}x\right)^{2}.

$$\left(-\frac{5}{3}\right)^{2}x^{2}=\left(-\sqrt{x+22}\right)^{2}$$

Calculate -\frac{5}{3} to the power of 2 and get \frac{25}{9}.

$$\frac{25}{9}x^{2}=\left(-\sqrt{x+22}\right)^{2}$$

Expand \left(-\sqrt{x+22}\right)^{2}.

$$\frac{25}{9}x^{2}=\left(-1\right)^{2}\left(\sqrt{x+22}\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$\frac{25}{9}x^{2}=1\left(\sqrt{x+22}\right)^{2}$$

Calculate \sqrt{x+22} to the power of 2 and get x+22.

$$\frac{25}{9}x^{2}=1\left(x+22\right)$$

Use the distributive property to multiply 1 by x+22.

$$\frac{25}{9}x^{2}=x+22$$

Subtract x from both sides.

$$\frac{25}{9}x^{2}-x=22$$

Subtract 22 from both sides.

$$\frac{25}{9}x^{2}-x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{25}{9} for a, -1 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{25}{9}\left(-22\right)}}{2\times \frac{25}{9}}$$

Multiply -4 times \frac{25}{9}.

$$x=\frac{-\left(-1\right)±\sqrt{1-\frac{100}{9}\left(-22\right)}}{2\times \frac{25}{9}}$$

Multiply -\frac{100}{9} times -22.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{2200}{9}}}{2\times \frac{25}{9}}$$

Add 1 to \frac{2200}{9}.

$$x=\frac{-\left(-1\right)±\sqrt{\frac{2209}{9}}}{2\times \frac{25}{9}}$$

Take the square root of \frac{2209}{9}.

$$x=\frac{-\left(-1\right)±\frac{47}{3}}{2\times \frac{25}{9}}$$

The opposite of -1 is 1.

$$x=\frac{1±\frac{47}{3}}{2\times \frac{25}{9}}$$

Multiply 2 times \frac{25}{9}.

$$x=\frac{1±\frac{47}{3}}{\frac{50}{9}}$$

Now solve the equation x=\frac{1±\frac{47}{3}}{\frac{50}{9}} when ± is plus. Add 1 to \frac{47}{3}.

$$x=\frac{\frac{50}{3}}{\frac{50}{9}}$$

Divide \frac{50}{3} by \frac{50}{9} by multiplying \frac{50}{3} by the reciprocal of \frac{50}{9}.

$$x=3$$

Now solve the equation x=\frac{1±\frac{47}{3}}{\frac{50}{9}} when ± is minus. Subtract \frac{47}{3} from 1.

$$x=-\frac{\frac{44}{3}}{\frac{50}{9}}$$

Divide -\frac{44}{3} by \frac{50}{9} by multiplying -\frac{44}{3} by the reciprocal of \frac{50}{9}.

$$x=-\frac{66}{25}$$

The equation is now solved.

$$x=3 x=-\frac{66}{25}$$

Substitute 3 for x in the equation -\frac{5}{3}x+2=5-\sqrt{x+22}-3.

$$-\frac{5}{3}\times 3+2=5-\sqrt{3+22}-3$$

Simplify. The value x=3 satisfies the equation.

$$-3=-3$$

Substitute -\frac{66}{25} for x in the equation -\frac{5}{3}x+2=5-\sqrt{x+22}-3.

$$-\frac{5}{3}\left(-\frac{66}{25}\right)+2=5-\sqrt{-\frac{66}{25}+22}-3$$

Simplify. The value x=-\frac{66}{25} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{32}{5}=-\frac{12}{5}$$

Equation -\frac{5x}{3}=-\sqrt{x+22} has a unique solution.

$$x=3$$