Solve x-2sqrt{5x}=15

$$x - 2 \sqrt { 5 x } = 15$$

Solve for x

$$x=45$$

Solution Steps

Subtract x from both sides of the equation.

$$-2\sqrt{5x}=15-x$$

Square both sides of the equation.

$$\left(-2\sqrt{5x}\right)^{2}=\left(15-x\right)^{2}$$

Expand \left(-2\sqrt{5x}\right)^{2}.

$$\left(-2\right)^{2}\left(\sqrt{5x}\right)^{2}=\left(15-x\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4\left(\sqrt{5x}\right)^{2}=\left(15-x\right)^{2}$$

Calculate \sqrt{5x} to the power of 2 and get 5x.

$$4\times 5x=\left(15-x\right)^{2}$$

Multiply 4 and 5 to get 20.

$$20x=\left(15-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(15-x\right)^{2}.

$$20x=225-30x+x^{2}$$

Subtract 225 from both sides.

$$20x-225=-30x+x^{2}$$

Add 30x to both sides.

$$20x-225+30x=x^{2}$$

Combine 20x and 30x to get 50x.

$$50x-225=x^{2}$$

Subtract x^{2} from both sides.

$$50x-225-x^{2}=0$$

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

$$-x^{2}+50x-225=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-225. To find a and b, set up a system to be solved.

$$a+b=50 ab=-\left(-225\right)=225$$

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 225.

$$1,225 3,75 5,45 9,25 15,15$$

Calculate the sum for each pair.

$$1+225=226 3+75=78 5+45=50 9+25=34 15+15=30$$

The solution is the pair that gives sum 50.

$$a=45 b=5$$

Rewrite -x^{2}+50x-225 as \left(-x^{2}+45x\right)+\left(5x-225\right).

$$\left(-x^{2}+45x\right)+\left(5x-225\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-45\right)+5\left(x-45\right)$$

Factor out common term x-45 by using distributive property.

$$\left(x-45\right)\left(-x+5\right)$$

To find equation solutions, solve x-45=0 and -x+5=0.

$$x=45 x=5$$

Substitute 45 for x in the equation x-2\sqrt{5x}=15.

$$45-2\sqrt{5\times 45}=15$$

Simplify. The value x=45 satisfies the equation.

$$15=15$$

Substitute 5 for x in the equation x-2\sqrt{5x}=15.

$$5-2\sqrt{5\times 5}=15$$

Simplify. The value x=5 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-5=15$$

Equation -2\sqrt{5x}=15-x has a unique solution.

$$x=45$$