Solve sqrt{3x+100}-x=4

$$\sqrt { 3 x + 100 } - x = 4$$

Solve for x

$$x=7$$

Solution Steps

Subtract -x from both sides of the equation.

$$\sqrt{3x+100}=4+x$$

Square both sides of the equation.

$$\left(\sqrt{3x+100}\right)^{2}=\left(4+x\right)^{2}$$

Calculate \sqrt{3x+100} to the power of 2 and get 3x+100.

$$3x+100=\left(4+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+x\right)^{2}.

$$3x+100=16+8x+x^{2}$$

Subtract 16 from both sides.

$$3x+100-16=8x+x^{2}$$

Subtract 16 from 100 to get 84.

$$3x+84=8x+x^{2}$$

Subtract 8x from both sides.

$$3x+84-8x=x^{2}$$

Combine 3x and -8x to get -5x.

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

Subtract x^{2} from both sides.

$$-5x+84-x^{2}=0$$

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

$$-x^{2}-5x+84=0$$

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.

$$a+b=-5 ab=-84=-84$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the 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.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

The solution is the pair that gives sum -5.

$$a=7 b=-12$$

Rewrite -x^{2}-5x+84 as \left(-x^{2}+7x\right)+\left(-12x+84\right).

$$\left(-x^{2}+7x\right)+\left(-12x+84\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(-x+7\right)+12\left(-x+7\right)$$

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

$$\left(-x+7\right)\left(x+12\right)$$

To find equation solutions, solve -x+7=0 and x+12=0.

$$x=7 x=-12$$

Substitute 7 for x in the equation \sqrt{3x+100}-x=4.

$$\sqrt{3\times 7+100}-7=4$$

Simplify. The value x=7 satisfies the equation.

$$4=4$$

Substitute -12 for x in the equation \sqrt{3x+100}-x=4.

$$\sqrt{3\left(-12\right)+100}-\left(-12\right)=4$$

Simplify. The value x=-12 does not satisfy the equation.

$$20=4$$

Equation \sqrt{3x+100}=x+4 has a unique solution.

$$x=7$$