Solve Quadratic Equations by Completing the Square: example 10.23

Solve $n^{2} = 3 n + 11$ by completing the square.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve $p^{2} = 5 p + 9$ by completing the square.

How did it go?


Subtract 3n to get the variable terms on the left side.
Take half of $− 3$ and square it. $(\frac{1}{2} (− 3))^{2} = \frac{9}{4}$
Add $\frac{9}{4}$ to both sides.
Factor the perfect square trinomial as a binomial square.
Add the fractions on the right side.
Use the Square Root Property.
Simplify the radical.
Solve for n.
Rewrite to show two equations.
Check. We leave the check for you!


Subtract 3n to get the variable terms on the left side.
Take half of $− 3$ and square it. $(\frac{1}{2} (− 3))^{2} = \frac{9}{4}$
Add $\frac{9}{4}$ to both sides.
Factor the perfect square trinomial as a binomial square.
Add the fractions on the right side.
Use the Square Root Property.
Simplify the radical.
Solve for n.
Rewrite to show two equations.
Check. We leave the check for you!