(x-6).jpeg "(x-6)(x-6)")
Understanding (x-6)(x-6)
The expression (x-6)(x-6) is a quadratic expression that represents the product of two identical binomials. Let's break down the steps to understand how to solve it and its implications.
Expanding the Expression
To simplify the expression, we use the distributive property. This means multiplying each term in the first binomial by each term in the second binomial:
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Step 1: (x - 6) * (x - 6) = x * (x - 6) - 6 * (x - 6)
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Step 2: Expanding further: x * x - x * 6 - 6 * x + 6 * 6
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Step 3: Simplifying: x² - 6x - 6x + 36
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Step 4: Combining like terms: x² - 12x + 36
Therefore, the expanded form of (x-6)(x-6) is x² - 12x + 36.
The Significance of Squaring a Binomial
Notice that the expression (x-6)(x-6) is essentially the square of the binomial (x-6). This means that the expanded form is a perfect square trinomial. Perfect square trinomials have a specific pattern that makes them easily recognizable:
- The first term is the square of the first term in the binomial (x² in this case).
- The last term is the square of the second term in the binomial (36 in this case).
- The middle term is twice the product of the two terms in the binomial (-12x in this case).
Applications
Understanding this expression and its expansion can be applied in various areas, including:
- Algebraic simplification: You can use this knowledge to simplify more complex algebraic expressions involving similar binomials.
- Factoring: The expanded form x² - 12x + 36 can be factored back into (x-6)(x-6). This is useful for solving quadratic equations.
- Geometry: If we consider x as a side length, the expression represents the area of a square with sides of length (x-6).
In Conclusion
(x-6)(x-6) is a simple but powerful expression that highlights the importance of understanding algebraic operations and their applications. By mastering this concept, you can tackle more complex problems in various mathematical contexts.