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Expanding (x+5)(x+5)
This expression represents the product of two identical binomials, (x+5) and (x+5). We can expand it using the FOIL method, which stands for First, Outer, Inner, Last.
Here's how it works:
1. First: Multiply the first terms of each binomial: x * x = x²
2. Outer: Multiply the outer terms of the binomials: x * 5 = 5x
3. Inner: Multiply the inner terms of the binomials: 5 * x = 5x
4. Last: Multiply the last terms of each binomial: 5 * 5 = 25
5. Combine Like Terms: Now, add all the terms together: x² + 5x + 5x + 25 = x² + 10x + 25
Therefore, the expanded form of (x+5)(x+5) is x² + 10x + 25.
Understanding the Result
The expanded form represents a quadratic expression, which is a polynomial with the highest power of the variable being 2. It can be visualized as a parabola when graphed.
This specific expression, x² + 10x + 25, is a perfect square trinomial. This means it can be factored back into the original form, (x+5)(x+5).
Applications
Expanding binomials like (x+5)(x+5) is essential in various mathematical contexts, including:
- Algebraic manipulation: Simplifying expressions, solving equations, and working with polynomials.
- Calculus: Finding derivatives and integrals.
- Geometry: Calculating areas and volumes.
- Physics and engineering: Modeling real-world phenomena.
Understanding how to expand binomials is a fundamental skill in mathematics.