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

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

Solve for x

$$x=1$$

Solution Steps

Subtract x-3 from both sides of the equation.

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

To find the opposite of x-3, find the opposite of each term.

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

The opposite of -3 is 3.

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

Square both sides of the equation.

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

Expand \left(2\sqrt{x}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

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

$$4x=\left(-x+3\right)^{2}$$

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

$$4x=x^{2}-6x+9$$

Subtract x^{2} from both sides.

$$4x-x^{2}=-6x+9$$

Add 6x to both sides.

$$4x-x^{2}+6x=9$$

Combine 4x and 6x to get 10x.

$$10x-x^{2}=9$$

Subtract 9 from both sides.

$$10x-x^{2}-9=0$$

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

$$-x^{2}+10x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=10 ab=-\left(-9\right)=9$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 9.

$$1,9 3,3$$

Calculate the sum for each pair.

$$1+9=10 3+3=6$$

The solution is the pair that gives sum 10.

$$a=9 b=1$$

Rewrite -x^{2}+10x-9 as \left(-x^{2}+9x\right)+\left(x-9\right).

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

Factor out -x in -x^{2}+9x.

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

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

$$\left(x-9\right)\left(-x+1\right)$$

To find equation solutions, solve x-9=0 and -x+1=0.

$$x=9 x=1$$

Substitute 9 for x in the equation x+2\sqrt{x}-3=0.

$$9+2\sqrt{9}-3=0$$

Simplify. The value x=9 does not satisfy the equation.

$$12=0$$

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

$$1+2\sqrt{1}-3=0$$

Simplify. The value x=1 satisfies the equation.

$$0=0$$

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

$$x=1$$