Solve sqrt{6x+13}=3+x

$$\sqrt { 6 x + 13 } = 3 + x$$

Solve for x

$$x=2$$

$$x=-2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{6x+13}\right)^{2}=\left(3+x\right)^{2}$$

Calculate \sqrt{6x+13} to the power of 2 and get 6x+13.

$$6x+13=\left(3+x\right)^{2}$$

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

$$6x+13=9+6x+x^{2}$$

Subtract 9 from both sides.

$$6x+13-9=6x+x^{2}$$

Subtract 9 from 13 to get 4.

$$6x+4=6x+x^{2}$$

Subtract 6x from both sides.

$$6x+4-6x=x^{2}$$

Combine 6x and -6x to get 0.

$$4=x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}=4$$

Subtract 4 from both sides.

$$x^{2}-4=0$$

Consider x^{2}-4. Rewrite x^{2}-4 as x^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-2\right)\left(x+2\right)=0$$

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

$$x=2 x=-2$$

Substitute 2 for x in the equation \sqrt{6x+13}=3+x.

$$\sqrt{6\times 2+13}=3+2$$

Simplify. The value x=2 satisfies the equation.

$$5=5$$

Substitute -2 for x in the equation \sqrt{6x+13}=3+x.

$$\sqrt{6\left(-2\right)+13}=3-2$$

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

$$1=1$$

List all solutions of \sqrt{6x+13}=x+3.

$$x=2 x=-2$$