Lesson 58: Completing the Square (Exploration: Modeling Completing the Square)

New Concept

Given a quadratic of the form $x^2 + bx$, add to it the square of half the coefficient of $x$, $(\frac{b}{2})^2$, to create a perfect square trinomial.

Why it matters

Completing the square is a powerful algebraic technique that transforms complex quadratic equations into a simpler, more insightful form. Mastering this method is crucial for understanding conic sections and for deriving the quadratic formula itself, a cornerstone of higher mathematics.

What’s next

Next, you’ll apply this method to solve quadratic equations that cannot be easily factored and see how it works in real-world problems.

Property

If $x^2 = a$, then $x = \pm\sqrt{a}$ for any $a > 0$.

Examples

Given a quadratic of the form $x^2 + bx$, add the square of half the coefficient of $x$, $\left(\frac{b}{2}\right)^2$, to create a perfect square trinomial.

Complete the square for $x^2 + 14x$. Add $\left(\frac{14}{2}\right)^2 = 7^2 = 49$. So, $x^2 + 14x + 49 = (x+7)^2$.
Complete the square for $y^2 - 20y$. Add $\left(\frac{-20}{2}\right)^2 = (-10)^2 = 100$. So, $y^2 - 20y + 100 = (y-10)^2$.

Imagine you have an incomplete puzzle, $x^2 + bx$. To turn it into a neat, perfect square, you need one special piece. This method is the secret to finding it! Just take the coefficient of $x$, which is 'b', chop it in half, and square the result. Adding this magic number completes the puzzle, making your expression perfectly factorable and easy to solve.

This method transforms a quadratic equation into a form that can be solved using the Square Root Property. The process involves isolating the variable terms, adding a constant to both sides to form a perfect square trinomial, factoring it, and then taking the square root of both sides to find the solutions for the variable.

Solve $x^2 - 8x - 9 = 0$. First, $x^2 - 8x = 9$. Add $(\frac{-8}{2})^2=16$ to both sides: $x^2 - 8x + 16 = 9 + 16$. This gives $(x-4)^2 = 25$, so $x-4 = \pm 5$, and $x=9$ or $x=-1$.
Solve $x^2 + 6x - 2 = 0$. First, $x^2 + 6x = 2$. Add $(\frac{6}{2})^2=9$ to both sides: $x^2 + 6x + 9 = 2 + 9$. This gives $(x+3)^2 = 11$, so $x = -3 \pm \sqrt{11}$.

This is your go-to strategy when a quadratic equation refuses to be factored easily. First, you shove the constant term to the other side. Then, you work your magic by 'completing the square' on the variable side, making sure to add the same value to the other side to keep the equation balanced. Finally, you use the Square Root Property to win the game.