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

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

Solve for x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Solution Steps

Subtract 2 from both sides of the equation.

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

Subtract 2 from -1 to get -3.

$$\sqrt{3x-1}=3x-3$$

Square both sides of the equation.

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

Calculate \sqrt{3x-1} to the power of 2 and get 3x-1.

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

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

$$3x-1=9x^{2}-18x+9$$

Subtract 9x^{2} from both sides.

$$3x-1-9x^{2}=-18x+9$$

Add 18x to both sides.

$$3x-1-9x^{2}+18x=9$$

Combine 3x and 18x to get 21x.

$$21x-1-9x^{2}=9$$

Subtract 9 from both sides.

$$21x-1-9x^{2}-9=0$$

Subtract 9 from -1 to get -10.

$$21x-10-9x^{2}=0$$

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

$$-9x^{2}+21x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=21 ab=-9\left(-10\right)=90$$

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 90.

$$1,90 2,45 3,30 5,18 6,15 9,10$$

Calculate the sum for each pair.

$$1+90=91 2+45=47 3+30=33 5+18=23 6+15=21 9+10=19$$

The solution is the pair that gives sum 21.

$$a=15 b=6$$

Rewrite -9x^{2}+21x-10 as \left(-9x^{2}+15x\right)+\left(6x-10\right).

$$\left(-9x^{2}+15x\right)+\left(6x-10\right)$$

Factor out -3x in the first and 2 in the second group.

$$-3x\left(3x-5\right)+2\left(3x-5\right)$$

Factor out common term 3x-5 by using distributive property.

$$\left(3x-5\right)\left(-3x+2\right)$$

To find equation solutions, solve 3x-5=0 and -3x+2=0.

$$x=\frac{5}{3} x=\frac{2}{3}$$

Substitute \frac{5}{3} for x in the equation 2+\sqrt{3x-1}=3x-1.

$$2+\sqrt{3\times \frac{5}{3}-1}=3\times \frac{5}{3}-1$$

Simplify. The value x=\frac{5}{3} satisfies the equation.

$$4=4$$

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

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

Simplify. The value x=\frac{2}{3} does not satisfy the equation.

$$3=1$$

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

$$x=\frac{5}{3}$$