Solve sqrt{7x+2}+x=6

$$\sqrt { 7 x + 2 } + x = 6$$

Solve for x

$$x=2$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{7x+2}=6-x$$

Square both sides of the equation.

$$\left(\sqrt{7x+2}\right)^{2}=\left(6-x\right)^{2}$$

Calculate \sqrt{7x+2} to the power of 2 and get 7x+2.

$$7x+2=\left(6-x\right)^{2}$$

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

$$7x+2=36-12x+x^{2}$$

Subtract 36 from both sides.

$$7x+2-36=-12x+x^{2}$$

Subtract 36 from 2 to get -34.

$$7x-34=-12x+x^{2}$$

Add 12x to both sides.

$$7x-34+12x=x^{2}$$

Combine 7x and 12x to get 19x.

$$19x-34=x^{2}$$

Subtract x^{2} from both sides.

$$19x-34-x^{2}=0$$

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

$$-x^{2}+19x-34=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-34. To find a and b, set up a system to be solved.

$$a+b=19 ab=-\left(-34\right)=34$$

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 34.

$$1,34 2,17$$

Calculate the sum for each pair.

$$1+34=35 2+17=19$$

The solution is the pair that gives sum 19.

$$a=17 b=2$$

Rewrite -x^{2}+19x-34 as \left(-x^{2}+17x\right)+\left(2x-34\right).

$$\left(-x^{2}+17x\right)+\left(2x-34\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-17\right)+2\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(-x+2\right)$$

To find equation solutions, solve x-17=0 and -x+2=0.

$$x=17 x=2$$

Substitute 17 for x in the equation \sqrt{7x+2}+x=6.

$$\sqrt{7\times 17+2}+17=6$$

Simplify. The value x=17 does not satisfy the equation.

$$28=6$$

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

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

Simplify. The value x=2 satisfies the equation.

$$6=6$$

Equation \sqrt{7x+2}=6-x has a unique solution.

$$x=2$$