(x%2B5).jpeg "(x-7)(x+5)")
Expanding and Understanding (x-7)(x+5)
This expression represents the product of two binomials: (x-7) and (x+5). To understand what this means and to simplify it, we can use the distributive property or the FOIL method.
Using the Distributive Property
The distributive property states that for any numbers a, b, and c:
- a(b + c) = ab + ac
We can apply this to our expression:
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First, treat (x+5) as a single entity and distribute (x-7) over it: (x-7)(x+5) = x(x+5) - 7(x+5)
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Now, distribute again within each term: x(x+5) - 7(x+5) = xx + x5 - 7x - 75
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Simplify by multiplying and combining like terms: x² + 5x - 7x - 35 = x² - 2x - 35
Therefore, (x-7)(x+5) expands to x² - 2x - 35.
Using the FOIL Method
FOIL stands for First, Outer, Inner, Last, and it provides a systematic way to multiply binomials.
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First: Multiply the first terms of each binomial: x * x = x²
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Outer: Multiply the outer terms: x * 5 = 5x
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Inner: Multiply the inner terms: -7 * x = -7x
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Last: Multiply the last terms: -7 * 5 = -35
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Combine the terms: x² + 5x - 7x - 35 = x² - 2x - 35
Again, we arrive at x² - 2x - 35.
What does this expression represent?
The expanded form x² - 2x - 35 represents a quadratic equation. It can be used to model various real-world scenarios involving quadratic relationships, such as projectile motion, area calculations, or growth patterns.
This expression can also be factored back into its original form, (x-7)(x+5), which is often useful for solving equations and finding roots.
Summary
Expanding (x-7)(x+5) using the distributive property or FOIL method results in x² - 2x - 35. This expression represents a quadratic equation with potential applications in various fields. Understanding how to expand and factor these expressions is crucial for solving problems in algebra and beyond.