Solve x-sqrt{x}=20

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

Solve for x

$$x=25$$

Solution Steps

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

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

$$\left(-1\right)^{2}\left(\sqrt{x}\right)^{2}=\left(20-x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{x}\right)^{2}=\left(20-x\right)^{2}$$

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

$$1x=\left(20-x\right)^{2}$$

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

$$1x=400-40x+x^{2}$$

Reorder the terms.

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

Subtract x^{2} from both sides.

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

Add 40x to both sides.

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

Combine x and 40x to get 41x.

$$41x-x^{2}=400$$

Subtract 400 from both sides.

$$41x-x^{2}-400=0$$

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

$$-x^{2}+41x-400=0$$

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.

$$a+b=41 ab=-\left(-400\right)=400$$

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 41.

$$a=25 b=16$$

Rewrite -x^{2}+41x-400 as \left(-x^{2}+25x\right)+\left(16x-400\right).

$$\left(-x^{2}+25x\right)+\left(16x-400\right)$$

Factor out -x in the first and 16 in the second group.

$$-x\left(x-25\right)+16\left(x-25\right)$$

Factor out common term x-25 by using distributive property.

$$\left(x-25\right)\left(-x+16\right)$$

To find equation solutions, solve x-25=0 and -x+16=0.

$$x=25 x=16$$

Substitute 25 for x in the equation x-\sqrt{x}=20.

$$25-\sqrt{25}=20$$

Simplify. The value x=25 satisfies the equation.

$$20=20$$

Substitute 16 for x in the equation x-\sqrt{x}=20.

$$16-\sqrt{16}=20$$

Simplify. The value x=16 does not satisfy the equation.

$$12=20$$

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

$$x=25$$