Solve p-2sqrt{p}=8

$$p - 2 \sqrt { p } = 8$$

Solve for p

$$p=16$$

Solution Steps

Subtract p from both sides of the equation.

$$-2\sqrt{p}=8-p$$

Square both sides of the equation.

$$\left(-2\sqrt{p}\right)^{2}=\left(8-p\right)^{2}$$

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

$$\left(-2\right)^{2}\left(\sqrt{p}\right)^{2}=\left(8-p\right)^{2}$$

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

$$4\left(\sqrt{p}\right)^{2}=\left(8-p\right)^{2}$$

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

$$4p=\left(8-p\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-p\right)^{2}.

$$4p=64-16p+p^{2}$$

Subtract 64 from both sides.

$$4p-64=-16p+p^{2}$$

Add 16p to both sides.

$$4p-64+16p=p^{2}$$

Combine 4p and 16p to get 20p.

$$20p-64=p^{2}$$

Subtract p^{2} from both sides.

$$20p-64-p^{2}=0$$

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

$$-p^{2}+20p-64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -p^{2}+ap+bp-64. To find a and b, set up a system to be solved.

$$a+b=20 ab=-\left(-64\right)=64$$

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 64.

$$1,64 2,32 4,16 8,8$$

Calculate the sum for each pair.

$$1+64=65 2+32=34 4+16=20 8+8=16$$

The solution is the pair that gives sum 20.

$$a=16 b=4$$

Rewrite -p^{2}+20p-64 as \left(-p^{2}+16p\right)+\left(4p-64\right).

$$\left(-p^{2}+16p\right)+\left(4p-64\right)$$

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

$$-p\left(p-16\right)+4\left(p-16\right)$$

Factor out common term p-16 by using distributive property.

$$\left(p-16\right)\left(-p+4\right)$$

To find equation solutions, solve p-16=0 and -p+4=0.

$$p=16 p=4$$

Substitute 16 for p in the equation p-2\sqrt{p}=8.

$$16-2\sqrt{16}=8$$

Simplify. The value p=16 satisfies the equation.

$$8=8$$

Substitute 4 for p in the equation p-2\sqrt{p}=8.

$$4-2\sqrt{4}=8$$

Simplify. The value p=4 does not satisfy the equation.

$$0=8$$

Equation -2\sqrt{p}=8-p has a unique solution.

$$p=16$$