Solve 3x=sqrt{11x-2}

$$3 x = \sqrt { 11 x - 2 }$$

Solve for x

$$x=1$$

$$x=\frac{2}{9}\approx 0.222222222$$

Solution Steps

Square both sides of the equation.

$$\left(3x\right)^{2}=\left(\sqrt{11x-2}\right)^{2}$$

Expand \left(3x\right)^{2}.

$$3^{2}x^{2}=\left(\sqrt{11x-2}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9x^{2}=\left(\sqrt{11x-2}\right)^{2}$$

Calculate \sqrt{11x-2} to the power of 2 and get 11x-2.

$$9x^{2}=11x-2$$

Subtract 11x from both sides.

$$9x^{2}-11x=-2$$

Add 2 to both sides.

$$9x^{2}-11x+2=0$$

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

$$a+b=-11 ab=9\times 2=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -11.

$$a=-9 b=-2$$

Rewrite 9x^{2}-11x+2 as \left(9x^{2}-9x\right)+\left(-2x+2\right).

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

Factor out 9x in the first and -2 in the second group.

$$9x\left(x-1\right)-2\left(x-1\right)$$

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

$$\left(x-1\right)\left(9x-2\right)$$

To find equation solutions, solve x-1=0 and 9x-2=0.

$$x=1 x=\frac{2}{9}$$

Substitute 1 for x in the equation 3x=\sqrt{11x-2}.

$$3\times 1=\sqrt{11\times 1-2}$$

Simplify. The value x=1 satisfies the equation.

$$3=3$$

Substitute \frac{2}{9} for x in the equation 3x=\sqrt{11x-2}.

$$3\times \frac{2}{9}=\sqrt{11\times \frac{2}{9}-2}$$

Simplify. The value x=\frac{2}{9} satisfies the equation.

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

List all solutions of 3x=\sqrt{11x-2}.

$$x=1 x=\frac{2}{9}$$