Blog Post 4: Learning to Complete the Square

Introduction
Hello everyone.  My name is Brenna, and today I will be teaching you how to complete the square.  By learning this concept, you will learn how to rewrite quadratic functions into vertex form and find the vertex of the function.

Review
As you have already learned, the standard form for a quadratic function is y = ax²  + bx + c, where a, b, and c are constants, and a ≠ 0.  Additionally, the parabola will open upward if a > 0.  The parabola will open downward if a < 0.  The parabola intersects the y-axis at c. 


The Vertex Form
With some quadratic functions, it is more useful to write the function in vertex form, such as when we want to find the maximum or minimum value of a parabola. 

Vertex Form:  y = a(x-h)² + k

When we write the function in vertex form, the parabola has vertex (h, k) and an axis of symmetry at x = h. 

Completing the Square

To convert from standard form to vertex form, we multiply out the squared term, which is known as completing the square.  Follow the example to learn how. 

Example: f(x) = x² + 6x + 2  

Step 1: Halve the b-term, then square it. 

            6/2 = 3 →  3² = 9

Step 2: Create a zero in the equation using the value from Step 1. 

            f(x) = x² + 6x + 9 - 9 + 2

Step 3: Rewrite the equation with a perfect square. 

            f(x) = (x+3)² -7

In this example, our vertex is (-3, -7).  Since our a-value is greater than one, the parabola opens up.  This means our vertex is the minimum value.  The axis of symmetry is at x = -3. 

When we graph the function, we can see the vertex and the axis of symmetry.

** Before you can complete the square, make sure that a = 1.  Factor if necessary. **

Geometric Visualization

If you find the concept difficult to understand, watch this video for a visual, geometric explanation.  (It also shows why we call it completing the "square"). 

Application
The flight path of an aircraft used to simulate weightlessness can be approximated by the equation h = 10t² + 300t + 9750, where h = height in meters and t = time in seconds.  What is the plane's maximum altitude?  How long does it take to reach the maximum altitude?

Step 1: Factor out -10 so that a = 0.
             -10(t² + 30t) + 9750

Step 2: Halve the b-value, then square it. 
             30/2 = 15  →  15² = 225

Step 3: Complete the square.
             -10(t² + 30t + 225 - 225) + 9750
             -10(t² + 30t + 225) + 2250 + 9750
             -10(t-15)²  + 12000

Step 4: Find maximum altitude and time.
             Vertex: (15, 12000) → Vertex at (15 seconds, 12000 m)
             Maximum altitude is 12000 m. It takes 15 seconds to reach this height.

Conclusion

As we can see, knowing how to complete the square allows you to manipulate quadratic functions and find important values, such as the minimum and maximum values of parabolas. These values have real life applications as well.