Solve x+sqrt{3x+10}=6

$$x + \sqrt { 3 x + 10 } = 6$$

Solve for x

$$x=2$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{3x+10}=6-x$$

Square both sides of the equation.

$$\left(\sqrt{3x+10}\right)^{2}=\left(6-x\right)^{2}$$

Calculate \sqrt{3x+10} to the power of 2 and get 3x+10.

$$3x+10=\left(6-x\right)^{2}$$

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

$$3x+10=36-12x+x^{2}$$

Subtract 36 from both sides.

$$3x+10-36=-12x+x^{2}$$

Subtract 36 from 10 to get -26.

$$3x-26=-12x+x^{2}$$

Add 12x to both sides.

$$3x-26+12x=x^{2}$$

Combine 3x and 12x to get 15x.

$$15x-26=x^{2}$$

Subtract x^{2} from both sides.

$$15x-26-x^{2}=0$$

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

$$-x^{2}+15x-26=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-26. To find a and b, set up a system to be solved.

$$a+b=15 ab=-\left(-26\right)=26$$

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 26.

$$1,26 2,13$$

Calculate the sum for each pair.

$$1+26=27 2+13=15$$

The solution is the pair that gives sum 15.

$$a=13 b=2$$

Rewrite -x^{2}+15x-26 as \left(-x^{2}+13x\right)+\left(2x-26\right).

$$\left(-x^{2}+13x\right)+\left(2x-26\right)$$

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

$$-x\left(x-13\right)+2\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(-x+2\right)$$

To find equation solutions, solve x-13=0 and -x+2=0.

$$x=13 x=2$$

Substitute 13 for x in the equation x+\sqrt{3x+10}=6.

$$13+\sqrt{3\times 13+10}=6$$

Simplify. The value x=13 does not satisfy the equation.

$$20=6$$

Substitute 2 for x in the equation x+\sqrt{3x+10}=6.

$$2+\sqrt{3\times 2+10}=6$$

Simplify. The value x=2 satisfies the equation.

$$6=6$$

Equation \sqrt{3x+10}=6-x has a unique solution.

$$x=2$$