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Completing the square is a fundamental technique in algebra, especially when dealing with quadratic equations. It’s a powerful tool that allows us to transform a quadratic equation into a more manageable form, making it easier to solve. In this comprehensive guide, we will explore the step-by-step process of completing the square, discuss its significance, and provide examples to illustrate its application.
What is Completing the Square?
Completing the square is a method used to rewrite a quadratic equation in the form of
(x + d)² + e = 0, where
e are constants. This form, known as a perfect square trinomial, simplifies the equation, making it easier to find the solutions for
x. Let’s dive into the process.
Step 1: Understanding the Basics
Before we begin, let’s clarify some essential terms and concepts:
- Quadratic Equation: A quadratic equation is any equation that can be written in the form
ax² + bx + c = 0, where
care constants, and
ais not equal to 0.
- Perfect Square Trinomial: A perfect square trinomial is an expression of the form
(x + d)², where
dis a constant.
Step 2: The Initial Equation
Consider the quadratic equation
ax² + bx + c = 0. To complete the square, we need to ensure that
a is equal to 1. If it’s not, we must divide the entire equation by
a to achieve this form:
x² + (b/a)x + (c/a) = 0.
This step simplifies the equation, making it easier to work with.
Step 3: Half of
Now, let’s focus on the middle term
(b/a)x. To complete the square, we need to add and subtract
(b/2a)². This term,
(b/2a)², is crucial for completing the square, as it ensures that the expression becomes a perfect square trinomial.
Step 4: Rearranging
Rearrange the equation to group the perfect square trinomial and the constants together:
x² + (b/a)x + (b/2a)² - (b/2a)² + (c/a) = 0.
Step 5: Simplify
Now, simplify the equation:
(x + (b/2a))² - (b/2a)² + (c/a) = 0.
Step 6: Isolate the Perfect Square Trinomial
Isolate the perfect square trinomial on the left side of the equation:
(x + (b/2a))² = (b/2a)² - (c/a).
Step 7: Taking the Square Root
Take the square root of both sides of the equation:
x + (b/2a) = ±√((b/2a)² - (c/a)).
Step 8: Isolate
x by subtracting
(b/2a) from both sides:
x = - (b/2a) ± √((b/2a)² - (c/a)).
Completing the Square in Action
Let’s apply the completing the square method to solve a quadratic equation:
x² + 4x + 1 = 0.
Step 1: Ensure that
a is equal to 1. In this case,
a = 1, so we can proceed.
Step 2: The equation is already in the simplified form.
Step 3: Find
(b/2a)² = (4/2)² = 2² = 4.
Step 4: Rearrange the equation:
x² + 4x + 4 - 4 + 1 = 0.
Step 5: Simplify:
(x + 2)² - 3 = 0.
Step 6: Isolate the perfect square trinomial:
(x + 2)² = 3.
Step 7: Take the square root:
x + 2 = ±√3.
Step 8: Isolate
x = -2 ± √3.
In this example, we have successfully completed the square and found the solutions for
x. They are
x = -2 + √3 and
x = -2 - √3.
Significance of Completing the Square
Completing the square is a versatile technique with several practical applications, making it a valuable tool in algebra and mathematics:
- Vertex Form: Completing the square allows us to express quadratic functions in vertex form,
(x - h)² + k, where
(h, k)represents the vertex of the parabola. This form provides valuable information about the graph of the quadratic equation.
- Solving Quadratic Equations: As demonstrated in our examples, completing the square is a reliable method for solving quadratic equations. It provides an alternative approach to the quadratic formula and is especially useful when the coefficients are not easily factorable.
- Geometric Interpretation: Completing the square has a geometric interpretation. It helps us visualize the transformation of a quadratic equation into a perfect square trinomial, making it easier to understand the behavior of quadratic functions.
- Derivation of the Quadratic Formula: Completing the square is an essential step in deriving the quadratic formula, which is a fundamental tool for solving quadratic equations of any form.
Completing the Square for General Quadratic Equations
In some cases, quadratic equations may have a coefficient
a in front of
x², as in
ax² + bx + c = 0. To complete the square for such equations, follow these steps:
Divide all terms by
a to achieve the simplified form:
x² + (b/a)x + (c/a) = 0.
Proceed with the completing the square method as outlined earlier.
Completing the square is a valuable technique that simplifies quadratic equations, making them easier to solve and providing insights into the behavior of quadratic functions. By following the step-by-step process and understanding its significance, you can confidently tackle quadratic equations, whether you’re finding solutions, graphing parabolas, or deriving important mathematical formulas. It’s a fundamental tool in the world of algebra, opening up a path to better solutions and a deeper understanding of mathematics.
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