Solve 2x+sqrt{6x+1}=3

$$2x+ \sqrt{ 6x+1 } =3$$

Solve for x

$$x=\frac{1}{2}=0.5$$

Solution Steps

Subtract 2x from both sides of the equation.

$$\sqrt{6x+1}=3-2x$$

Square both sides of the equation.

$$\left(\sqrt{6x+1}\right)^{2}=\left(3-2x\right)^{2}$$

Calculate \sqrt{6x+1} to the power of 2 and get 6x+1.

$$6x+1=\left(3-2x\right)^{2}$$

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

$$6x+1=9-12x+4x^{2}$$

Subtract 9 from both sides.

$$6x+1-9=-12x+4x^{2}$$

Subtract 9 from 1 to get -8.

$$6x-8=-12x+4x^{2}$$

Add 12x to both sides.

$$6x-8+12x=4x^{2}$$

Combine 6x and 12x to get 18x.

$$18x-8=4x^{2}$$

Subtract 4x^{2} from both sides.

$$18x-8-4x^{2}=0$$

Divide both sides by 2.

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

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

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

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

$$a+b=9 ab=-2\left(-4\right)=8$$

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 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 9.

$$a=8 b=1$$

Rewrite -2x^{2}+9x-4 as \left(-2x^{2}+8x\right)+\left(x-4\right).

$$\left(-2x^{2}+8x\right)+\left(x-4\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(-x+4\right)-\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(2x-1\right)$$

To find equation solutions, solve -x+4=0 and 2x-1=0.

$$x=4 x=\frac{1}{2}$$

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

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

Simplify. The value x=4 does not satisfy the equation.

$$13=3$$

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

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

Simplify. The value x=\frac{1}{2} satisfies the equation.

$$3=3$$

Equation \sqrt{6x+1}=3-2x has a unique solution.

$$x=\frac{1}{2}$$