Solve sqrt{5x+16}+x=10

$$\sqrt { 5 x + 16 } + x = 10$$

Solve for x

$$x=4$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{5x+16}=10-x$$

Square both sides of the equation.

$$\left(\sqrt{5x+16}\right)^{2}=\left(10-x\right)^{2}$$

Calculate \sqrt{5x+16} to the power of 2 and get 5x+16.

$$5x+16=\left(10-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-x\right)^{2}.

$$5x+16=100-20x+x^{2}$$

Subtract 100 from both sides.

$$5x+16-100=-20x+x^{2}$$

Subtract 100 from 16 to get -84.

$$5x-84=-20x+x^{2}$$

Add 20x to both sides.

$$5x-84+20x=x^{2}$$

Combine 5x and 20x to get 25x.

$$25x-84=x^{2}$$

Subtract x^{2} from both sides.

$$25x-84-x^{2}=0$$

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

$$-x^{2}+25x-84=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-84. To find a and b, set up a system to be solved.

$$a+b=25 ab=-\left(-84\right)=84$$

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 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Calculate the sum for each pair.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

The solution is the pair that gives sum 25.

$$a=21 b=4$$

Rewrite -x^{2}+25x-84 as \left(-x^{2}+21x\right)+\left(4x-84\right).

$$\left(-x^{2}+21x\right)+\left(4x-84\right)$$

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

$$-x\left(x-21\right)+4\left(x-21\right)$$

Factor out common term x-21 by using distributive property.

$$\left(x-21\right)\left(-x+4\right)$$

To find equation solutions, solve x-21=0 and -x+4=0.

$$x=21 x=4$$

Substitute 21 for x in the equation \sqrt{5x+16}+x=10.

$$\sqrt{5\times 21+16}+21=10$$

Simplify. The value x=21 does not satisfy the equation.

$$32=10$$

Substitute 4 for x in the equation \sqrt{5x+16}+x=10.

$$\sqrt{5\times 4+16}+4=10$$

Simplify. The value x=4 satisfies the equation.

$$10=10$$

Equation \sqrt{5x+16}=10-x has a unique solution.

$$x=4$$