Solve x-sqrt{2x+3}=0

$$x - \sqrt { 2 x + 3 } = 0$$

Solve for x

$$x=3$$

Solution Steps

Subtract x from both sides of the equation.

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

Cancel out -1 on both sides.

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

Square both sides of the equation.

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

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

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

Subtract x^{2} from both sides.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

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

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.

$$a=3 b=-1$$

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

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

Factor out -x in the first and -1 in the second group.

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

Factor out common term x-3 by using distributive property.

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

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

$$x=3 x=-1$$

Substitute 3 for x in the equation x-\sqrt{2x+3}=0.

$$3-\sqrt{2\times 3+3}=0$$

Simplify. The value x=3 satisfies the equation.

$$0=0$$

Substitute -1 for x in the equation x-\sqrt{2x+3}=0.

$$-1-\sqrt{2\left(-1\right)+3}=0$$

Simplify. The value x=-1 does not satisfy the equation.

$$-2=0$$

Equation \sqrt{2x+3}=x has a unique solution.

$$x=3$$