Solve x-3sqrt{x-5}=5

$$x - 3 \sqrt { x - 5 } = 5$$

Solve for x

$$x=14$$

$$x=5$$

Solution Steps

Subtract x from both sides of the equation.

$$-3\sqrt{x-5}=5-x$$

Square both sides of the equation.

$$\left(-3\sqrt{x-5}\right)^{2}=\left(5-x\right)^{2}$$

Expand \left(-3\sqrt{x-5}\right)^{2}.

$$\left(-3\right)^{2}\left(\sqrt{x-5}\right)^{2}=\left(5-x\right)^{2}$$

Calculate -3 to the power of 2 and get 9.

$$9\left(\sqrt{x-5}\right)^{2}=\left(5-x\right)^{2}$$

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

$$9\left(x-5\right)=\left(5-x\right)^{2}$$

Use the distributive property to multiply 9 by x-5.

$$9x-45=\left(5-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$9x-45=25-10x+x^{2}$$

Subtract 25 from both sides.

$$9x-45-25=-10x+x^{2}$$

Subtract 25 from -45 to get -70.

$$9x-70=-10x+x^{2}$$

Add 10x to both sides.

$$9x-70+10x=x^{2}$$

Combine 9x and 10x to get 19x.

$$19x-70=x^{2}$$

Subtract x^{2} from both sides.

$$19x-70-x^{2}=0$$

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

$$-x^{2}+19x-70=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-70. To find a and b, set up a system to be solved.

$$a+b=19 ab=-\left(-70\right)=70$$

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 70.

$$1,70 2,35 5,14 7,10$$

Calculate the sum for each pair.

$$1+70=71 2+35=37 5+14=19 7+10=17$$

The solution is the pair that gives sum 19.

$$a=14 b=5$$

Rewrite -x^{2}+19x-70 as \left(-x^{2}+14x\right)+\left(5x-70\right).

$$\left(-x^{2}+14x\right)+\left(5x-70\right)$$

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

$$-x\left(x-14\right)+5\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(-x+5\right)$$

To find equation solutions, solve x-14=0 and -x+5=0.

$$x=14 x=5$$

Substitute 14 for x in the equation x-3\sqrt{x-5}=5.

$$14-3\sqrt{14-5}=5$$

Simplify. The value x=14 satisfies the equation.

$$5=5$$

Substitute 5 for x in the equation x-3\sqrt{x-5}=5.

$$5-3\sqrt{5-5}=5$$

Simplify. The value x=5 satisfies the equation.

$$5=5$$

List all solutions of -3\sqrt{x-5}=5-x.

$$x=14 x=5$$