Solve sqrt{3x+7}-3=x

$$\sqrt { 3 x + 7 } - 3 = x$$

Solve for x

$$x=-1$$

$$x=-2$$

Solution Steps

Subtract -3 from both sides of the equation.

$$\sqrt{3x+7}=x+3$$

Square both sides of the equation.

$$\left(\sqrt{3x+7}\right)^{2}=\left(x+3\right)^{2}$$

Calculate \sqrt{3x+7} to the power of 2 and get 3x+7.

$$3x+7=\left(x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$3x+7=x^{2}+6x+9$$

Subtract x^{2} from both sides.

$$3x+7-x^{2}=6x+9$$

Subtract 6x from both sides.

$$3x+7-x^{2}-6x=9$$

Combine 3x and -6x to get -3x.

$$-3x+7-x^{2}=9$$

Subtract 9 from both sides.

$$-3x+7-x^{2}-9=0$$

Subtract 9 from 7 to get -2.

$$-3x-2-x^{2}=0$$

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

$$-x^{2}-3x-2=0$$

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

$$a+b=-3 ab=-\left(-2\right)=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-2$$

Rewrite -x^{2}-3x-2 as \left(-x^{2}-x\right)+\left(-2x-2\right).

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

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

$$x\left(-x-1\right)+2\left(-x-1\right)$$

Factor out common term -x-1 by using distributive property.

$$\left(-x-1\right)\left(x+2\right)$$

To find equation solutions, solve -x-1=0 and x+2=0.

$$x=-1 x=-2$$

Substitute -1 for x in the equation \sqrt{3x+7}-3=x.

$$\sqrt{3\left(-1\right)+7}-3=-1$$

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

$$-1=-1$$

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

$$\sqrt{3\left(-2\right)+7}-3=-2$$

Simplify. The value x=-2 satisfies the equation.

$$-2=-2$$

List all solutions of \sqrt{3x+7}=x+3.

$$x=-1 x=-2$$