Simplifying and Analyzing the Quadratic Expression 3x^2 - 12x - -12
Introduction:
The expression 3x^2 - 12x - -12 is a quadratic expression involving a single variable, x. In this article, we will explore the process of simplifying and analyzing this expression. We will discuss factoring, identifying roots, and graphing to understand its behavior fully.
Steps to Simplify the Expression
Let’s start by simplifying the given expression:
Step 1: Rewrite the expression without the double negative sign. Step 2: Factor out the greatest common factor. Step 3: Further factorize the quadratic expression.1. Initial Expression
We begin with the expression:
3x^2 - 12x - -12
Since negative signs can be confusing, let's rewrite it more clearly:
3x^2 - 12x 12
2. Factoring Out the Greatest Common Factor
The next step is to factor out the greatest common factor (GCF) of the terms:
3x^2 - 12x 12 3(x^2 - 4x 4)
3. Further Factoring the Quadratic Expression
Now, we factorize the quadratic expression inside the parentheses:
x^2 - 4x 4 (x - 2)^2
Therefore, the simplified expression is:
3(x - 2)^2
Analysis of the Expression
Let’s discuss the significance and implications of this simplified expression:
1. Graphing the Expression
The expression can be rewritten as:
y 3(x - 2)^2
This is a parabolic equation with the vertex at (2, 0).
Graph: [Graph image here]
2. Identifying Roots and X-Intercepts
Roots or x-intercepts are the values of x where y 0:
3(x - 2)^2 0
(x - 2)^2 0
x - 2 0
x 2
So, the expression has a double root at x 2.
3. Behavior at Different Values of x
For all other real values of x, the expression will yield a positive result:
When x 2, y > 0
When x > 2, y > 0
The parabola opens upwards, meaning that the function values increase sharply as x moves away from 2 in either direction.
Understanding Quadratic Expressions
A quadratic expression in a single variable is generally expressed as:
ax^2 bx c
In our case:
a 3, b -12, c 12
This type of expression is essential in various fields, including physics, engineering, and economics, where understanding the behavior of quadratic functions is crucial.
Conclusion
In conclusion, the expression 3x^2 - 12x - -12 simplifies to 3(x - 2)^2. This expression represents a parabola with a vertex at (2, 0) and opens upwards. The double root at x 2 indicates the vertex of the parabola, and the expression is positive for all other real values of x.
Understanding and analyzing quadratic expressions is a fundamental skill in algebra and has numerous practical applications.
Related Keywords: quadratic expression, simplification, algebraic expression