Completing the square is a crucial algebraic technique used to solve quadratic equations, rewrite quadratic functions in vertex form, and simplify various mathematical expressions. While it might seem daunting at first, with a little practice, it becomes a straightforward process. This guide will walk you through the steps, providing clear explanations and examples.
Understanding the Concept
Before diving into the process, let's understand the underlying concept. Completing the square involves manipulating a quadratic expression of the form ax² + bx + c to create a perfect square trinomial—a trinomial that can be factored into the square of a binomial, (x + p)² or (x - p)². This perfect square trinomial is then easily solvable.
Steps to Complete the Square
The steps to complete the square depend on whether the coefficient of the x² term (a) is 1 or not.
Case 1: Coefficient of x² is 1 (a = 1)
Let's consider a quadratic expression of the form x² + bx + c. Here's how to complete the square:
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Identify 'b': Determine the coefficient of the x term (b).
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Find (b/2)²: Divide 'b' by 2 and square the result. This is the crucial step that creates the perfect square trinomial.
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Add and Subtract (b/2)²: Add and subtract (b/2)² to the expression. This doesn't change the value of the expression because you're essentially adding zero.
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Factor the Perfect Square Trinomial: The first three terms will now form a perfect square trinomial, which can be factored into the square of a binomial:
(x + b/2)². -
Simplify: Simplify the remaining constant term.
Example: Complete the square for x² + 6x + 2.
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b = 6
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(b/2)² = (6/2)² = 9
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x² + 6x + 9 - 9 + 2
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(x + 3)² - 7
Therefore, completing the square for x² + 6x + 2 gives us (x + 3)² - 7.
Case 2: Coefficient of x² is NOT 1 (a ≠ 1)
When the coefficient of x² is not 1, an additional step is required before proceeding as in Case 1.
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Factor out 'a': Factor out the coefficient of x² from the x² and x terms.
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Complete the Square (as in Case 1): Follow steps 1-5 from Case 1 for the expression within the parentheses.
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Distribute 'a': Distribute the factored-out 'a' back into the expression.
Example: Complete the square for 2x² + 8x + 5.
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Factor out 2: 2(x² + 4x) + 5
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Complete the square within the parentheses: b = 4, (b/2)² = 4. So we have 2(x² + 4x + 4 - 4) + 5
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Simplify and Distribute: 2((x + 2)² - 4) + 5 = 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3
Therefore, completing the square for 2x² + 8x + 5 results in 2(x + 2)² - 3.
Applications of Completing the Square
Completing the square has numerous applications in algebra and beyond:
- Solving Quadratic Equations: By completing the square, you can easily solve quadratic equations that are not easily factorable.
- Finding the Vertex of a Parabola: Completing the square transforms a quadratic function into its vertex form,
a(x - h)² + k, where (h, k) represents the vertex of the parabola. - Graphing Quadratic Functions: The vertex form obtained through completing the square helps in accurately graphing quadratic functions.
- Calculus: It simplifies certain integral calculations.
Mastering the technique of completing the square is a valuable skill for anyone studying algebra and related fields. Practice with various examples to build your proficiency and confidence. Remember, the key is to understand the underlying principle of creating a perfect square trinomial and then systematically following the steps.
