Solve x+2=sqrt{2x+7}

$$x + 2 = \sqrt { 2 x + 7 }$$

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

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

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

$$x^{2}+4x+4=\left(\sqrt{2x+7}\right)^{2}$$

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

$$x^{2}+4x+4=2x+7$$

Subtract 2x from both sides.

$$x^{2}+4x+4-2x=7$$

Combine 4x and -2x to get 2x.

$$x^{2}+2x+4=7$$

Subtract 7 from both sides.

$$x^{2}+2x+4-7=0$$

Subtract 7 from 4 to get -3.

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

To solve the equation, factor x^{2}+2x-3 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=2 ab=-3$$

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. The only such pair is the system solution.

$$a=-1 b=3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-1\right)\left(x+3\right)$$

To find equation solutions, solve x-1=0 and x+3=0.

$$x=1 x=-3$$

Substitute 1 for x in the equation x+2=\sqrt{2x+7}.

$$1+2=\sqrt{2\times 1+7}$$

Simplify. The value x=1 satisfies the equation.

$$3=3$$

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

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

Simplify. The value x=-3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation x+2=\sqrt{2x+7} has a unique solution.

$$x=1$$