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

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

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$x+3=9x^{2}-6x+1$$

Subtract 9x^{2} from both sides.

$$x+3-9x^{2}=-6x+1$$

Add 6x to both sides.

$$x+3-9x^{2}+6x=1$$

Combine x and 6x to get 7x.

$$7x+3-9x^{2}=1$$

Subtract 1 from both sides.

$$7x+3-9x^{2}-1=0$$

Subtract 1 from 3 to get 2.

$$7x+2-9x^{2}=0$$

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

$$-9x^{2}+7x+2=0$$

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.

$$a+b=7 ab=-9\times 2=-18$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -18.

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

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 7.

$$a=9 b=-2$$

Rewrite -9x^{2}+7x+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 \sqrt{x+3}=3x-1.

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

Simplify. The value x=1 satisfies the equation.

$$2=2$$

Substitute -\frac{2}{9} for x in the equation \sqrt{x+3}=3x-1.

$$\sqrt{-\frac{2}{9}+3}=3\left(-\frac{2}{9}\right)-1$$

Simplify. The value x=-\frac{2}{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+3}=3x-1 has a unique solution.

$$x=1$$