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Completing the Square: A Step-by-Step Guide
Completing the square is a powerful technique used to solve quadratic equations. It allows us to rewrite the equation in a form where we can easily find the solutions. While the quadratic formula can be used to solve any quadratic equation, completing the square offers a deeper understanding of the process and can be helpful in other areas of mathematics.
Understanding the Process
The core idea behind completing the square is to manipulate a quadratic equation of the form ax² + bx + c = 0 into a perfect square trinomial on one side of the equation. A perfect square trinomial is a trinomial that can be factored as (x + h)² or (x - h)².
Steps to Completing the Square
Let's break down the process with a step-by-step example:
1. Move the Constant Term
Begin by moving the constant term (c) to the right side of the equation. For example, if we have the equation x² + 6x + 5 = 0, we would rewrite it as x² + 6x = -5.
2. Complete the Square
To complete the square, we need to find a constant value that, when added to both sides of the equation, will create a perfect square trinomial on the left side. We achieve this by taking half of the coefficient of the x term (b), squaring it, and adding it to both sides. In our example, the coefficient of the x term is 6. Half of 6 is 3, and squaring 3 gives us 9. Adding 9 to both sides, we get:
x² + 6x + 9 = -5 + 9
The left side of the equation now forms a perfect square trinomial: (x + 3)².
3. Factor the Perfect Square
Factor the perfect square trinomial: (x + 3)² = 4
4. Apply the Square Root Property
Apply the square root property to both sides of the equation. This means taking the square root of both sides. Remember to include both positive and negative roots:
√(x + 3)² = ±√4
Simplify: x + 3 = ±2
5. Solve for x
Isolate x by subtracting 3 from both sides:
x = -3 ± 2
Therefore, the solutions to the equation x² + 6x + 5 = 0 are x = -1 and x = -5.
Key Points to Remember
- If the coefficient of the x² term is not 1, divide the entire equation by that coefficient before completing the square.
- Completing the square can be used to solve quadratic equations, find the vertex of a parabola, and solve other types of equations that involve squares.
- Practice with various examples to solidify your understanding of the process.