Solve p-2sqrt{p}=15

$$p - 2 \sqrt { p } = 15$$

Solve for p

$$p=25$$

Solution Steps

Subtract p from both sides of the equation.

$$-2\sqrt{p}=15-p$$

Square both sides of the equation.

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

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

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

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

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

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

$$4p=\left(15-p\right)^{2}$$

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

$$4p=225-30p+p^{2}$$

Subtract 225 from both sides.

$$4p-225=-30p+p^{2}$$

Add 30p to both sides.

$$4p-225+30p=p^{2}$$

Combine 4p and 30p to get 34p.

$$34p-225=p^{2}$$

Subtract p^{2} from both sides.

$$34p-225-p^{2}=0$$

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

$$-p^{2}+34p-225=0$$

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

$$a+b=34 ab=-\left(-225\right)=225$$

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 34.

$$a=25 b=9$$

Rewrite -p^{2}+34p-225 as \left(-p^{2}+25p\right)+\left(9p-225\right).

$$\left(-p^{2}+25p\right)+\left(9p-225\right)$$

Factor out -p in the first and 9 in the second group.

$$-p\left(p-25\right)+9\left(p-25\right)$$

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

$$\left(p-25\right)\left(-p+9\right)$$

To find equation solutions, solve p-25=0 and -p+9=0.

$$p=25 p=9$$

Substitute 25 for p in the equation p-2\sqrt{p}=15.

$$25-2\sqrt{25}=15$$

Simplify. The value p=25 satisfies the equation.

$$15=15$$

Substitute 9 for p in the equation p-2\sqrt{p}=15.

$$9-2\sqrt{9}=15$$

Simplify. The value p=9 does not satisfy the equation.

$$3=15$$

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

$$p=25$$