Solve x-8sqrt{x}=20

$$x-8 \sqrt{ x } =20$$

Solve for x

$$x=100$$

Solution Steps

Subtract x from both sides of the equation.

$$-8\sqrt{x}=20-x$$

Square both sides of the equation.

$$\left(-8\sqrt{x}\right)^{2}=\left(20-x\right)^{2}$$

Expand \left(-8\sqrt{x}\right)^{2}.

$$\left(-8\right)^{2}\left(\sqrt{x}\right)^{2}=\left(20-x\right)^{2}$$

Calculate -8 to the power of 2 and get 64.

$$64\left(\sqrt{x}\right)^{2}=\left(20-x\right)^{2}$$

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

$$64x=\left(20-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(20-x\right)^{2}.

$$64x=400-40x+x^{2}$$

Subtract 400 from both sides.

$$64x-400=-40x+x^{2}$$

Add 40x to both sides.

$$64x-400+40x=x^{2}$$

Combine 64x and 40x to get 104x.

$$104x-400=x^{2}$$

Subtract x^{2} from both sides.

$$104x-400-x^{2}=0$$

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

$$-x^{2}+104x-400=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-400. To find a and b, set up a system to be solved.

$$a+b=104 ab=-\left(-400\right)=400$$

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 400.

$$1,400 2,200 4,100 5,80 8,50 10,40 16,25 20,20$$

Calculate the sum for each pair.

$$1+400=401 2+200=202 4+100=104 5+80=85 8+50=58 10+40=50 16+25=41 20+20=40$$

The solution is the pair that gives sum 104.

$$a=100 b=4$$

Rewrite -x^{2}+104x-400 as \left(-x^{2}+100x\right)+\left(4x-400\right).

$$\left(-x^{2}+100x\right)+\left(4x-400\right)$$

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

$$-x\left(x-100\right)+4\left(x-100\right)$$

Factor out common term x-100 by using distributive property.

$$\left(x-100\right)\left(-x+4\right)$$

To find equation solutions, solve x-100=0 and -x+4=0.

$$x=100 x=4$$

Substitute 100 for x in the equation x-8\sqrt{x}=20.

$$100-8\sqrt{100}=20$$

Simplify. The value x=100 satisfies the equation.

$$20=20$$

Substitute 4 for x in the equation x-8\sqrt{x}=20.

$$4-8\sqrt{4}=20$$

Simplify. The value x=4 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-12=20$$

Equation -8\sqrt{x}=20-x has a unique solution.

$$x=100$$