Solve p=sqrt{10-9p}

$$p = \sqrt { 10 - 9 p }$$

Solve for p

$$p=1$$

Solution Steps

Square both sides of the equation.

$$p^{2}=\left(\sqrt{10-9p}\right)^{2}$$

Calculate \sqrt{10-9p} to the power of 2 and get 10-9p.

$$p^{2}=10-9p$$

Subtract 10 from both sides.

$$p^{2}-10=-9p$$

Add 9p to both sides.

$$p^{2}-10+9p=0$$

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

$$p^{2}+9p-10=0$$

To solve the equation, factor p^{2}+9p-10 using formula p^{2}+\left(a+b\right)p+ab=\left(p+a\right)\left(p+b\right). To find a and b, set up a system to be solved.

$$a+b=9 ab=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 9.

$$a=-1 b=10$$

Rewrite factored expression \left(p+a\right)\left(p+b\right) using the obtained values.

$$\left(p-1\right)\left(p+10\right)$$

To find equation solutions, solve p-1=0 and p+10=0.

$$p=1 p=-10$$

Substitute 1 for p in the equation p=\sqrt{10-9p}.

$$1=\sqrt{10-9}$$

Simplify. The value p=1 satisfies the equation.

$$1=1$$

Substitute -10 for p in the equation p=\sqrt{10-9p}.

$$-10=\sqrt{10-9\left(-10\right)}$$

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

$$-10=10$$

Equation p=\sqrt{10-9p} has a unique solution.

$$p=1$$