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

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

Solve for x

$$x = \frac{22}{9} = 2\frac{4}{9} \approx 2.444444444$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+3}\right)^{2}=\left(3x-5\right)^{2}$$

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

$$x+3=\left(3x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-5\right)^{2}.

$$x+3=9x^{2}-30x+25$$

Subtract 9x^{2} from both sides.

$$x+3-9x^{2}=-30x+25$$

Add 30x to both sides.

$$x+3-9x^{2}+30x=25$$

Combine x and 30x to get 31x.

$$31x+3-9x^{2}=25$$

Subtract 25 from both sides.

$$31x+3-9x^{2}-25=0$$

Subtract 25 from 3 to get -22.

$$31x-22-9x^{2}=0$$

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

$$-9x^{2}+31x-22=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-22. To find a and b, set up a system to be solved.

$$a+b=31 ab=-9\left(-22\right)=198$$

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 198.

$$1,198 2,99 3,66 6,33 9,22 11,18$$

Calculate the sum for each pair.

$$1+198=199 2+99=101 3+66=69 6+33=39 9+22=31 11+18=29$$

The solution is the pair that gives sum 31.

$$a=22 b=9$$

Rewrite -9x^{2}+31x-22 as \left(-9x^{2}+22x\right)+\left(9x-22\right).

$$\left(-9x^{2}+22x\right)+\left(9x-22\right)$$

Factor out -x in -9x^{2}+22x.

$$-x\left(9x-22\right)+9x-22$$

Factor out common term 9x-22 by using distributive property.

$$\left(9x-22\right)\left(-x+1\right)$$

To find equation solutions, solve 9x-22=0 and -x+1=0.

$$x=\frac{22}{9} x=1$$

Substitute \frac{22}{9} for x in the equation \sqrt{x+3}=3x-5.

$$\sqrt{\frac{22}{9}+3}=3\times \frac{22}{9}-5$$

Simplify. The value x=\frac{22}{9} satisfies the equation.

$$\frac{7}{3}=\frac{7}{3}$$

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

$$\sqrt{1+3}=3\times 1-5$$

Simplify. The value x=1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{x+3}=3x-5 has a unique solution.

$$x=\frac{22}{9}$$