Solve 3x=sqrt{8+6x}

$$3 x = \sqrt { 8 + 6 x }$$

Solve for x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Solution Steps

Square both sides of the equation.

$$\left(3x\right)^{2}=\left(\sqrt{8+6x}\right)^{2}$$

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

$$3^{2}x^{2}=\left(\sqrt{8+6x}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9x^{2}=\left(\sqrt{8+6x}\right)^{2}$$

Calculate \sqrt{8+6x} to the power of 2 and get 8+6x.

$$9x^{2}=8+6x$$

Subtract 8 from both sides.

$$9x^{2}-8=6x$$

Subtract 6x from both sides.

$$9x^{2}-8-6x=0$$

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

$$9x^{2}-6x-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=-6 ab=9\left(-8\right)=-72$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -6.

$$a=-12 b=6$$

Rewrite 9x^{2}-6x-8 as \left(9x^{2}-12x\right)+\left(6x-8\right).

$$\left(9x^{2}-12x\right)+\left(6x-8\right)$$

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

$$3x\left(3x-4\right)+2\left(3x-4\right)$$

Factor out common term 3x-4 by using distributive property.

$$\left(3x-4\right)\left(3x+2\right)$$

To find equation solutions, solve 3x-4=0 and 3x+2=0.

$$x=\frac{4}{3} x=-\frac{2}{3}$$

Substitute \frac{4}{3} for x in the equation 3x=\sqrt{8+6x}.

$$3\times \frac{4}{3}=\sqrt{8+6\times \frac{4}{3}}$$

Simplify. The value x=\frac{4}{3} satisfies the equation.

$$4=4$$

Substitute -\frac{2}{3} for x in the equation 3x=\sqrt{8+6x}.

$$3\left(-\frac{2}{3}\right)=\sqrt{8+6\left(-\frac{2}{3}\right)}$$

Simplify. The value x=-\frac{2}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Equation 3x=\sqrt{6x+8} has a unique solution.

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