Lesson 3: Completing the Square

Property

To make a perfect square trinomial from an expression like $x^2 + bx$, we use the binomial squares pattern. The goal is to find a number to add that completes the pattern $a^2 + 2ab + b^2 = (a+b)^2$.

To complete the square of $x^2 + bx$:
Step 1. Identify $b$, the coefficient of $x$.
Step 2. Find $(\frac{1}{2}b)^2$, the number to complete the square.
Step 3. Add the $(\frac{1}{2}b)^2$ to $x^2 + bx$. The result is $x^2 + bx + (\frac{1}{2}b)^2$, which factors to $(x + \frac{b}{2})^2$.

Examples

• To complete the square for $x^2 + 12x$, we find $(\frac{1}{2} \cdot 12)^2 = 6^2 = 36$. The perfect square trinomial is $x^2 + 12x + 36$, which factors to $(x+6)^2$.

Property

To solve a quadratic equation of the form $x^2+bx+c=0$:

1. Move the constant term to the other side: $x^2+bx = -c$.
2. Complete the square on the left. Add $p^2 = (\frac{b}{2})^2$ to both sides of the equation: $x^2+bx+p^2 = -c+p^2$.
3. Write the left side as a binomial squared: $(x+p)^2 = -c+p^2$.
4. Use extraction of roots to find the solutions.

Examples

• To solve $x^2 - 6x - 7 = 0$, first write $x^2-6x=7$. Add $(\frac{-6}{2})^2=9$ to both sides: $x^2-6x+9=7+9$, so $(x-3)^2=16$. The solutions are $x=7$ and $x=-1$.
• To solve $x^2 - 4x - 3 = 0$, first write $x^2-4x=3$. Add $(\frac{-4}{2})^2=4$ to both sides: $x^2-4x+4=3+4$, so $(x-2)^2=7$. The solutions are $x = 2 \pm \sqrt{7}$.
• To solve $x^2+9x+20=0$, first write $x^2+9x=-20$. Add $(\frac{9}{2})^2=\frac{81}{4}$ to both sides: $x^2+9x+\frac{81}{4}=-20+\frac{81}{4}$, so $(x+\frac{9}{2})^2=\frac{1}{4}$. The solutions are $x=-4$ and $x=-5$.

Explanation

This method transforms any quadratic equation into the form $(x+p)^2=q$. By forcing one side to be a perfect square, we can easily solve for $x$ by taking the square root of both sides, which is often simpler than factoring.