Solve x+sqrt{x}=30

$$x + \sqrt { x } = 30$$

Solve for x

$$x=25$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{x}=30-x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(30-x\right)^{2}$$

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

$$x=\left(30-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(30-x\right)^{2}.

$$x=900-60x+x^{2}$$

Subtract 900 from both sides.

$$x-900=-60x+x^{2}$$

Add 60x to both sides.

$$x-900+60x=x^{2}$$

Combine x and 60x to get 61x.

$$61x-900=x^{2}$$

Subtract x^{2} from both sides.

$$61x-900-x^{2}=0$$

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

$$-x^{2}+61x-900=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-900. To find a and b, set up a system to be solved.

$$a+b=61 ab=-\left(-900\right)=900$$

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 900.

$$1,900 2,450 3,300 4,225 5,180 6,150 9,100 10,90 12,75 15,60 18,50 20,45 25,36 30,30$$

Calculate the sum for each pair.

$$1+900=901 2+450=452 3+300=303 4+225=229 5+180=185 6+150=156 9+100=109 10+90=100 12+75=87 15+60=75 18+50=68 20+45=65 25+36=61 30+30=60$$

The solution is the pair that gives sum 61.

$$a=36 b=25$$

Rewrite -x^{2}+61x-900 as \left(-x^{2}+36x\right)+\left(25x-900\right).

$$\left(-x^{2}+36x\right)+\left(25x-900\right)$$

Factor out -x in the first and 25 in the second group.

$$-x\left(x-36\right)+25\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(-x+25\right)$$

To find equation solutions, solve x-36=0 and -x+25=0.

$$x=36 x=25$$

Substitute 36 for x in the equation x+\sqrt{x}=30.

$$36+\sqrt{36}=30$$

Simplify. The value x=36 does not satisfy the equation.

$$42=30$$

Substitute 25 for x in the equation x+\sqrt{x}=30.

$$25+\sqrt{25}=30$$

Simplify. The value x=25 satisfies the equation.

$$30=30$$

Equation \sqrt{x}=30-x has a unique solution.

$$x=25$$