Solve sqrt{1+9x}+6=2x

$$\sqrt { 1 + 9 x } + 6 = 2 x$$

Solve for x

$$x=7$$

Solution Steps

Subtract 6 from both sides of the equation.

$$\sqrt{1+9x}=2x-6$$

Square both sides of the equation.

$$\left(\sqrt{1+9x}\right)^{2}=\left(2x-6\right)^{2}$$

Calculate \sqrt{1+9x} to the power of 2 and get 1+9x.

$$1+9x=\left(2x-6\right)^{2}$$

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

$$1+9x=4x^{2}-24x+36$$

Subtract 4x^{2} from both sides.

$$1+9x-4x^{2}=-24x+36$$

Add 24x to both sides.

$$1+9x-4x^{2}+24x=36$$

Combine 9x and 24x to get 33x.

$$1+33x-4x^{2}=36$$

Subtract 36 from both sides.

$$1+33x-4x^{2}-36=0$$

Subtract 36 from 1 to get -35.

$$-35+33x-4x^{2}=0$$

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

$$-4x^{2}+33x-35=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-35. To find a and b, set up a system to be solved.

$$a+b=33 ab=-4\left(-35\right)=140$$

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 140.

$$1,140 2,70 4,35 5,28 7,20 10,14$$

Calculate the sum for each pair.

$$1+140=141 2+70=72 4+35=39 5+28=33 7+20=27 10+14=24$$

The solution is the pair that gives sum 33.

$$a=28 b=5$$

Rewrite -4x^{2}+33x-35 as \left(-4x^{2}+28x\right)+\left(5x-35\right).

$$\left(-4x^{2}+28x\right)+\left(5x-35\right)$$

Factor out 4x in the first and -5 in the second group.

$$4x\left(-x+7\right)-5\left(-x+7\right)$$

Factor out common term -x+7 by using distributive property.

$$\left(-x+7\right)\left(4x-5\right)$$

To find equation solutions, solve -x+7=0 and 4x-5=0.

$$x=7 x=\frac{5}{4}$$

Substitute 7 for x in the equation \sqrt{1+9x}+6=2x.

$$\sqrt{1+9\times 7}+6=2\times 7$$

Simplify. The value x=7 satisfies the equation.

$$14=14$$

Substitute \frac{5}{4} for x in the equation \sqrt{1+9x}+6=2x.

$$\sqrt{1+9\times \frac{5}{4}}+6=2\times \frac{5}{4}$$

Simplify. The value x=\frac{5}{4} does not satisfy the equation.

$$\frac{19}{2}=\frac{5}{2}$$

Equation \sqrt{9x+1}=2x-6 has a unique solution.

$$x=7$$