Solve p-4sqrt{p}=21

$$p - 4 \sqrt { p } = 21$$

Solve for p

$$p=49$$

Solution Steps

Subtract p from both sides of the equation.

$$-4\sqrt{p}=21-p$$

Square both sides of the equation.

$$\left(-4\sqrt{p}\right)^{2}=\left(21-p\right)^{2}$$

Expand \left(-4\sqrt{p}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{p}\right)^{2}=\left(21-p\right)^{2}$$

Calculate -4 to the power of 2 and get 16.

$$16\left(\sqrt{p}\right)^{2}=\left(21-p\right)^{2}$$

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

$$16p=\left(21-p\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(21-p\right)^{2}.

$$16p=441-42p+p^{2}$$

Subtract 441 from both sides.

$$16p-441=-42p+p^{2}$$

Add 42p to both sides.

$$16p-441+42p=p^{2}$$

Combine 16p and 42p to get 58p.

$$58p-441=p^{2}$$

Subtract p^{2} from both sides.

$$58p-441-p^{2}=0$$

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

$$-p^{2}+58p-441=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -p^{2}+ap+bp-441. To find a and b, set up a system to be solved.

$$a+b=58 ab=-\left(-441\right)=441$$

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 441.

$$1,441 3,147 7,63 9,49 21,21$$

Calculate the sum for each pair.

$$1+441=442 3+147=150 7+63=70 9+49=58 21+21=42$$

The solution is the pair that gives sum 58.

$$a=49 b=9$$

Rewrite -p^{2}+58p-441 as \left(-p^{2}+49p\right)+\left(9p-441\right).

$$\left(-p^{2}+49p\right)+\left(9p-441\right)$$

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

$$-p\left(p-49\right)+9\left(p-49\right)$$

Factor out common term p-49 by using distributive property.

$$\left(p-49\right)\left(-p+9\right)$$

To find equation solutions, solve p-49=0 and -p+9=0.

$$p=49 p=9$$

Substitute 49 for p in the equation p-4\sqrt{p}=21.

$$49-4\sqrt{49}=21$$

Simplify. The value p=49 satisfies the equation.

$$21=21$$

Substitute 9 for p in the equation p-4\sqrt{p}=21.

$$9-4\sqrt{9}=21$$

Simplify. The value p=9 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-3=21$$

Equation -4\sqrt{p}=21-p has a unique solution.

$$p=49$$