Solve sqrt{3x+15}+3=x+2

$$\sqrt { 3 x + 15 } + 3 = x + 2$$

Solve for x

$$x=7$$

Solution Steps

Subtract 3 from both sides of the equation.

$$\sqrt{3x+15}=x+2-3$$

Subtract 3 from 2 to get -1.

$$\sqrt{3x+15}=x-1$$

Square both sides of the equation.

$$\left(\sqrt{3x+15}\right)^{2}=\left(x-1\right)^{2}$$

Calculate \sqrt{3x+15} to the power of 2 and get 3x+15.

$$3x+15=\left(x-1\right)^{2}$$

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

$$3x+15=x^{2}-2x+1$$

Subtract x^{2} from both sides.

$$3x+15-x^{2}=-2x+1$$

Add 2x to both sides.

$$3x+15-x^{2}+2x=1$$

Combine 3x and 2x to get 5x.

$$5x+15-x^{2}=1$$

Subtract 1 from both sides.

$$5x+15-x^{2}-1=0$$

Subtract 1 from 15 to get 14.

$$5x+14-x^{2}=0$$

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

$$-x^{2}+5x+14=0$$

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.

$$a+b=5 ab=-14=-14$$

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 5.

$$a=7 b=-2$$

Rewrite -x^{2}+5x+14 as \left(-x^{2}+7x\right)+\left(-2x+14\right).

$$\left(-x^{2}+7x\right)+\left(-2x+14\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(x-7\right)-2\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x-2\right)$$

To find equation solutions, solve x-7=0 and -x-2=0.

$$x=7 x=-2$$

Substitute 7 for x in the equation \sqrt{3x+15}+3=x+2.

$$\sqrt{3\times 7+15}+3=7+2$$

Simplify. The value x=7 satisfies the equation.

$$9=9$$

Substitute -2 for x in the equation \sqrt{3x+15}+3=x+2.

$$\sqrt{3\left(-2\right)+15}+3=-2+2$$

Simplify. The value x=-2 does not satisfy the equation.

$$6=0$$

Equation \sqrt{3x+15}=x-1 has a unique solution.

$$x=7$$