Solve x+3sqrt{x}-10=0

$$x + 3 \sqrt { x } - 10 = 0$$

Solve for x

$$x=4$$

Solution Steps

Subtract x-10 from both sides of the equation.

$$3\sqrt{x}=-\left(x-10\right)$$

To find the opposite of x-10, find the opposite of each term.

$$3\sqrt{x}=-x-\left(-10\right)$$

The opposite of -10 is 10.

$$3\sqrt{x}=-x+10$$

Square both sides of the equation.

$$\left(3\sqrt{x}\right)^{2}=\left(-x+10\right)^{2}$$

Expand \left(3\sqrt{x}\right)^{2}.

$$3^{2}\left(\sqrt{x}\right)^{2}=\left(-x+10\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{x}\right)^{2}=\left(-x+10\right)^{2}$$

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

$$9x=\left(-x+10\right)^{2}$$

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

$$9x=x^{2}-20x+100$$

Subtract x^{2} from both sides.

$$9x-x^{2}=-20x+100$$

Add 20x to both sides.

$$9x-x^{2}+20x=100$$

Combine 9x and 20x to get 29x.

$$29x-x^{2}=100$$

Subtract 100 from both sides.

$$29x-x^{2}-100=0$$

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

$$-x^{2}+29x-100=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-100. To find a and b, set up a system to be solved.

$$a+b=29 ab=-\left(-100\right)=100$$

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 100.

$$1,100 2,50 4,25 5,20 10,10$$

Calculate the sum for each pair.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

The solution is the pair that gives sum 29.

$$a=25 b=4$$

Rewrite -x^{2}+29x-100 as \left(-x^{2}+25x\right)+\left(4x-100\right).

$$\left(-x^{2}+25x\right)+\left(4x-100\right)$$

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

$$-x\left(x-25\right)+4\left(x-25\right)$$

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

$$\left(x-25\right)\left(-x+4\right)$$

To find equation solutions, solve x-25=0 and -x+4=0.

$$x=25 x=4$$

Substitute 25 for x in the equation x+3\sqrt{x}-10=0.

$$25+3\sqrt{25}-10=0$$

Simplify. The value x=25 does not satisfy the equation.

$$30=0$$

Substitute 4 for x in the equation x+3\sqrt{x}-10=0.

$$4+3\sqrt{4}-10=0$$

Simplify. The value x=4 satisfies the equation.

$$0=0$$

Equation 3\sqrt{x}=10-x has a unique solution.

$$x=4$$