Solve 2x+sqrt{x+6}=3

$$2 x + \sqrt { x + 6 } = 3$$

Solve for x

$$x=\frac{1}{4}=0.25$$

Solution Steps

Subtract 2x from both sides of the equation.

$$\sqrt{x+6}=3-2x$$

Square both sides of the equation.

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

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

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

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

$$x+6=9-12x+4x^{2}$$

Subtract 9 from both sides.

$$x+6-9=-12x+4x^{2}$$

Subtract 9 from 6 to get -3.

$$x-3=-12x+4x^{2}$$

Add 12x to both sides.

$$x-3+12x=4x^{2}$$

Combine x and 12x to get 13x.

$$13x-3=4x^{2}$$

Subtract 4x^{2} from both sides.

$$13x-3-4x^{2}=0$$

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

$$-4x^{2}+13x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=13 ab=-4\left(-3\right)=12$$

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 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 13.

$$a=12 b=1$$

Rewrite -4x^{2}+13x-3 as \left(-4x^{2}+12x\right)+\left(x-3\right).

$$\left(-4x^{2}+12x\right)+\left(x-3\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(-x+3\right)-\left(-x+3\right)$$

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

$$\left(-x+3\right)\left(4x-1\right)$$

To find equation solutions, solve -x+3=0 and 4x-1=0.

$$x=3 x=\frac{1}{4}$$

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

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

Simplify. The value x=3 does not satisfy the equation.

$$9=3$$

Substitute \frac{1}{4} for x in the equation 2x+\sqrt{x+6}=3.

$$2\times \frac{1}{4}+\sqrt{\frac{1}{4}+6}=3$$

Simplify. The value x=\frac{1}{4} satisfies the equation.

$$3=3$$

Equation \sqrt{x+6}=3-2x has a unique solution.

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