Completing the square is a powerful technique in mathematics, essential for solving quadratic equations in E-Math, finding the turning point of quadratic graphs, and determining the center of a circle’s equation in A-Math. However, students often find this method challenging to remember. In this guide, we’ll break down the process of completing the square into simple steps, ensuring that you can easily recall and apply it in your math tuition classes.
What is Completing the Square?
Completing the square involves rewriting a quadratic expression in the form ax^2 + bx + c into the form a(x-h)^2 + k, making it easier to solve equations, analyze graphs, and understand geometric properties. This method is particularly useful in O-Levels Math for understanding quadratic functions and equations.
Fundamental Step: The Core Formula
The key to completing the square lies in this formula:
ax^2 + bx + c = a(x-h)^2 + k
To make this formula easier to understand, let’s go through the following examples and observe the pattern:
Example 1: x^2 + 6x
To complete the square, add and subtract (6/2)^2 = 9:
x^2 + 6x = (x + 3)^2 – 9
Example 2: x^2 – 4x
Add and subtract (-4/2)^2 = 4:
x^2 – 4x = (x – 2)^2 – 4
Application to Typical Quadratic Equations
To apply completing the square to a general quadratic equation with three terms, follow the same steps as above, but include the constant term. For example:
x^2 + 6x + 5
Start with x^2 + 6x and complete the square:
x^2 + 6x = (x + 3)^2 – 9
Add the constant term back:
(x + 3)^2 – 9 + 5 = (x + 3)^2 – 4
Note: Further simplify the equation after completing the square.
Handling More Complex Quadratic Equations
When the quadratic term x^2 has a coefficient other than 1, factor it out before completing the square. For example:
2x^2 + 8x + 6
Factor out the 2: 2(x^2 + 4x) + 6
Complete the square inside the parentheses:
2(x^2 + 4x) = 2((x + 2)^2 – 4)
Distribute and simplify:
2((x + 2)^2 – 4) + 6 = 2(x + 2)^2 – 8 + 6
2(x + 2)^2 – 2
Practice Makes Perfect
To master completing the square, practice is essential. The more you practice, the more intuitive the process becomes. Remember the core formula and practice with different types of quadratic equations to build confidence.
Conclusion
Completing the square is a fundamental technique in O-Levels Math. With a solid understanding of the core formula and plenty of practice, you’ll find it easy to solve quadratic equations and analyze quadratic graphs. Remember, “Don’t practice until you get it right, practice until you can’t get it wrong!” Keep practicing, and you’ll master this skill in no time.
