Solve x+3sqrt{x+2}=2

$$x + 3 \sqrt { x + 2 } = 2$$

Solve for x

$$x=-1$$

Solution Steps

Subtract x from both sides of the equation.

$$3\sqrt{x+2}=2-x$$

Square both sides of the equation.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Use the distributive property to multiply 9 by x+2.

$$9x+18=\left(2-x\right)^{2}$$

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

$$9x+18=4-4x+x^{2}$$

Subtract 4 from both sides.

$$9x+18-4=-4x+x^{2}$$

Subtract 4 from 18 to get 14.

$$9x+14=-4x+x^{2}$$

Add 4x to both sides.

$$9x+14+4x=x^{2}$$

Combine 9x and 4x to get 13x.

$$13x+14=x^{2}$$

Subtract x^{2} from both sides.

$$13x+14-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+13x+14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=13 ab=-14=-14$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -14.

$$-1,14 -2,7$$

Calculate the sum for each pair.

$$-1+14=13 -2+7=5$$

The solution is the pair that gives sum 13.

$$a=14 b=-1$$

Rewrite -x^{2}+13x+14 as \left(-x^{2}+14x\right)+\left(-x+14\right).

$$\left(-x^{2}+14x\right)+\left(-x+14\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(x-14\right)-\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(-x-1\right)$$

To find equation solutions, solve x-14=0 and -x-1=0.

$$x=14 x=-1$$

Substitute 14 for x in the equation x+3\sqrt{x+2}=2.

$$14+3\sqrt{14+2}=2$$

Simplify. The value x=14 does not satisfy the equation.

$$26=2$$

Substitute -1 for x in the equation x+3\sqrt{x+2}=2.

$$-1+3\sqrt{-1+2}=2$$

Simplify. The value x=-1 satisfies the equation.

$$2=2$$

Equation 3\sqrt{x+2}=2-x has a unique solution.

$$x=-1$$