Solve sqrt{x+2}=3x-4

$$\sqrt { x + 2 } = 3 x - 4$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+2}\right)^{2}=\left(3x-4\right)^{2}$$

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-4\right)^{2}.

$$x+2=9x^{2}-24x+16$$

Subtract 9x^{2} from both sides.

$$x+2-9x^{2}=-24x+16$$

Add 24x to both sides.

$$x+2-9x^{2}+24x=16$$

Combine x and 24x to get 25x.

$$25x+2-9x^{2}=16$$

Subtract 16 from both sides.

$$25x+2-9x^{2}-16=0$$

Subtract 16 from 2 to get -14.

$$25x-14-9x^{2}=0$$

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

$$-9x^{2}+25x-14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-14. To find a and b, set up a system to be solved.

$$a+b=25 ab=-9\left(-14\right)=126$$

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 126.

$$1,126 2,63 3,42 6,21 7,18 9,14$$

Calculate the sum for each pair.

$$1+126=127 2+63=65 3+42=45 6+21=27 7+18=25 9+14=23$$

The solution is the pair that gives sum 25.

$$a=18 b=7$$

Rewrite -9x^{2}+25x-14 as \left(-9x^{2}+18x\right)+\left(7x-14\right).

$$\left(-9x^{2}+18x\right)+\left(7x-14\right)$$

Factor out 9x in the first and -7 in the second group.

$$9x\left(-x+2\right)-7\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(9x-7\right)$$

To find equation solutions, solve -x+2=0 and 9x-7=0.

$$x=2 x=\frac{7}{9}$$

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

$$\sqrt{2+2}=3\times 2-4$$

Simplify. The value x=2 satisfies the equation.

$$2=2$$

Substitute \frac{7}{9} for x in the equation \sqrt{x+2}=3x-4.

$$\sqrt{\frac{7}{9}+2}=3\times \frac{7}{9}-4$$

Simplify. The value x=\frac{7}{9} does not satisfy the equation because the left and the right hand side have opposite signs.

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

Equation \sqrt{x+2}=3x-4 has a unique solution.

$$x=2$$