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

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

Solve for x

$$x=-1$$

Solution Steps

Subtract 2x+3 from both sides of the equation.

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

Cancel out -1 on both sides.

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

Square both sides of the equation.

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

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

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

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

$$-x=4x^{2}+12x+9$$

Subtract 4x^{2} from both sides.

$$-x-4x^{2}=12x+9$$

Subtract 12x from both sides.

$$-x-4x^{2}-12x=9$$

Subtract 9 from both sides.

$$-x-4x^{2}-12x-9=0$$

Combine -x and -12x to get -13x.

$$-13x-4x^{2}-9=0$$

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

$$-4x^{2}-13x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=-13 ab=-4\left(-9\right)=36$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -13.

$$a=-4 b=-9$$

Rewrite -4x^{2}-13x-9 as \left(-4x^{2}-4x\right)+\left(-9x-9\right).

$$\left(-4x^{2}-4x\right)+\left(-9x-9\right)$$

Factor out 4x in the first and 9 in the second group.

$$4x\left(-x-1\right)+9\left(-x-1\right)$$

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

$$\left(-x-1\right)\left(4x+9\right)$$

To find equation solutions, solve -x-1=0 and 4x+9=0.

$$x=-1 x=-\frac{9}{4}$$

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

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

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

$$0=0$$

Substitute -\frac{9}{4} for x in the equation 2x-\sqrt{-x}+3=0.

$$2\left(-\frac{9}{4}\right)-\sqrt{-\left(-\frac{9}{4}\right)}+3=0$$

Simplify. The value x=-\frac{9}{4} does not satisfy the equation.

$$-3=0$$

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

$$x=-1$$