Solve 3x+6=sqrt{22-9x}

$$3x+6= \sqrt{ 22-9x }$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

Solution Steps

Square both sides of the equation.

$$\left(3x+6\right)^{2}=\left(\sqrt{22-9x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+6\right)^{2}.

$$9x^{2}+36x+36=\left(\sqrt{22-9x}\right)^{2}$$

Calculate \sqrt{22-9x} to the power of 2 and get 22-9x.

$$9x^{2}+36x+36=22-9x$$

Subtract 22 from both sides.

$$9x^{2}+36x+36-22=-9x$$

Subtract 22 from 36 to get 14.

$$9x^{2}+36x+14=-9x$$

Add 9x to both sides.

$$9x^{2}+36x+14+9x=0$$

Combine 36x and 9x to get 45x.

$$9x^{2}+45x+14=0$$

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.

$$a+b=45 ab=9\times 14=126$$

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 45.

$$a=3 b=42$$

Rewrite 9x^{2}+45x+14 as \left(9x^{2}+3x\right)+\left(42x+14\right).

$$\left(9x^{2}+3x\right)+\left(42x+14\right)$$

Factor out 3x in the first and 14 in the second group.

$$3x\left(3x+1\right)+14\left(3x+1\right)$$

Factor out common term 3x+1 by using distributive property.

$$\left(3x+1\right)\left(3x+14\right)$$

To find equation solutions, solve 3x+1=0 and 3x+14=0.

$$x=-\frac{1}{3} x=-\frac{14}{3}$$

Substitute -\frac{1}{3} for x in the equation 3x+6=\sqrt{22-9x}.

$$3\left(-\frac{1}{3}\right)+6=\sqrt{22-9\left(-\frac{1}{3}\right)}$$

Simplify. The value x=-\frac{1}{3} satisfies the equation.

$$5=5$$

Substitute -\frac{14}{3} for x in the equation 3x+6=\sqrt{22-9x}.

$$3\left(-\frac{14}{3}\right)+6=\sqrt{22-9\left(-\frac{14}{3}\right)}$$

Simplify. The value x=-\frac{14}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-8=8$$

Equation 3x+6=\sqrt{22-9x} has a unique solution.

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