Solve sqrt{3x+37}+3=x+2

$$\sqrt { 3 x + 37 } + 3 = x + 2$$

Solve for x

$$x=9$$

Solution Steps

Subtract 3 from both sides of the equation.

$$\sqrt{3x+37}=x+2-3$$

Subtract 3 from 2 to get -1.

$$\sqrt{3x+37}=x-1$$

Square both sides of the equation.

$$\left(\sqrt{3x+37}\right)^{2}=\left(x-1\right)^{2}$$

Calculate \sqrt{3x+37} to the power of 2 and get 3x+37.

$$3x+37=\left(x-1\right)^{2}$$

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

$$3x+37=x^{2}-2x+1$$

Subtract x^{2} from both sides.

$$3x+37-x^{2}=-2x+1$$

Add 2x to both sides.

$$3x+37-x^{2}+2x=1$$

Combine 3x and 2x to get 5x.

$$5x+37-x^{2}=1$$

Subtract 1 from both sides.

$$5x+37-x^{2}-1=0$$

Subtract 1 from 37 to get 36.

$$5x+36-x^{2}=0$$

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

$$-x^{2}+5x+36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+36. To find a and b, set up a system to be solved.

$$a+b=5 ab=-36=-36$$

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 -36.

$$-1,36 -2,18 -3,12 -4,9 -6,6$$

Calculate the sum for each pair.

$$-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0$$

The solution is the pair that gives sum 5.

$$a=9 b=-4$$

Rewrite -x^{2}+5x+36 as \left(-x^{2}+9x\right)+\left(-4x+36\right).

$$\left(-x^{2}+9x\right)+\left(-4x+36\right)$$

Factor out -x in the first and -4 in the second group.

$$-x\left(x-9\right)-4\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(-x-4\right)$$

To find equation solutions, solve x-9=0 and -x-4=0.

$$x=9 x=-4$$

Substitute 9 for x in the equation \sqrt{3x+37}+3=x+2.

$$\sqrt{3\times 9+37}+3=9+2$$

Simplify. The value x=9 satisfies the equation.

$$11=11$$

Substitute -4 for x in the equation \sqrt{3x+37}+3=x+2.

$$\sqrt{3\left(-4\right)+37}+3=-4+2$$

Simplify. The value x=-4 does not satisfy the equation because the left and the right hand side have opposite signs.

$$8=-2$$

Equation \sqrt{3x+37}=x-1 has a unique solution.

$$x=9$$