Solve sqrt{90-p}=p

$$\sqrt { 90 - p } = p$$

Solve for p

$$p=9$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{90-p}\right)^{2}=p^{2}$$

Calculate \sqrt{90-p} to the power of 2 and get 90-p.

$$90-p=p^{2}$$

Subtract p^{2} from both sides.

$$90-p-p^{2}=0$$

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

$$-p^{2}-p+90=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -p^{2}+ap+bp+90. To find a and b, set up a system to be solved.

$$a+b=-1 ab=-90=-90$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Calculate the sum for each pair.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

The solution is the pair that gives sum -1.

$$a=9 b=-10$$

Rewrite -p^{2}-p+90 as \left(-p^{2}+9p\right)+\left(-10p+90\right).

$$\left(-p^{2}+9p\right)+\left(-10p+90\right)$$

Factor out p in the first and 10 in the second group.

$$p\left(-p+9\right)+10\left(-p+9\right)$$

Factor out common term -p+9 by using distributive property.

$$\left(-p+9\right)\left(p+10\right)$$

To find equation solutions, solve -p+9=0 and p+10=0.

$$p=9 p=-10$$

Substitute 9 for p in the equation \sqrt{90-p}=p.

$$\sqrt{90-9}=9$$

Simplify. The value p=9 satisfies the equation.

$$9=9$$

Substitute -10 for p in the equation \sqrt{90-p}=p.

$$\sqrt{90-\left(-10\right)}=-10$$

Simplify. The value p=-10 does not satisfy the equation because the left and the right hand side have opposite signs.

$$10=-10$$

Equation \sqrt{90-p}=p has a unique solution.

$$p=9$$