Solve sqrt{3x}+7=3x+5

$$\sqrt { 3 x } + 7 = 3 x + 5$$

Solve for x

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

Solution Steps

Subtract 7 from both sides of the equation.

$$\sqrt{3x}=3x+5-7$$

Subtract 7 from 5 to get -2.

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

Square both sides of the equation.

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

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

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

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

$$3x=9x^{2}-12x+4$$

Subtract 9x^{2} from both sides.

$$3x-9x^{2}=-12x+4$$

Add 12x to both sides.

$$3x-9x^{2}+12x=4$$

Combine 3x and 12x to get 15x.

$$15x-9x^{2}=4$$

Subtract 4 from both sides.

$$15x-9x^{2}-4=0$$

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

$$-9x^{2}+15x-4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=15 ab=-9\left(-4\right)=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 15.

$$a=12 b=3$$

Rewrite -9x^{2}+15x-4 as \left(-9x^{2}+12x\right)+\left(3x-4\right).

$$\left(-9x^{2}+12x\right)+\left(3x-4\right)$$

Factor out -3x in -9x^{2}+12x.

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

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

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

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

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

Substitute \frac{4}{3} for x in the equation \sqrt{3x}+7=3x+5.

$$\sqrt{3\times \frac{4}{3}}+7=3\times \frac{4}{3}+5$$

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

$$9=9$$

Substitute \frac{1}{3} for x in the equation \sqrt{3x}+7=3x+5.

$$\sqrt{3\times \frac{1}{3}}+7=3\times \frac{1}{3}+5$$

Simplify. The value x=\frac{1}{3} does not satisfy the equation.

$$8=6$$

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

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