(x-6).jpeg "(x-4)(x-6)")
Factoring and Expanding (x-4)(x-6)
This article will explore the concepts of factoring and expanding as they relate to the expression (x-4)(x-6).
Expanding the Expression
Expanding the expression means multiplying the terms inside the parentheses. We can use the FOIL method (First, Outer, Inner, Last) to do this:
- First: Multiply the first terms of each binomial: x * x = x²
- Outer: Multiply the outer terms of the binomials: x * -6 = -6x
- Inner: Multiply the inner terms of the binomials: -4 * x = -4x
- Last: Multiply the last terms of each binomial: -4 * -6 = 24
Now, combine the terms: x² - 6x - 4x + 24
Finally, simplify by combining like terms: x² - 10x + 24
Therefore, the expanded form of (x-4)(x-6) is x² - 10x + 24.
Factoring the Expression
Factoring is the reverse process of expanding. We're given a quadratic expression (x² - 10x + 24) and need to find the two binomials that multiply to give us the original expression.
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Find two numbers that add up to the coefficient of the x term (-10) and multiply to the constant term (24). The numbers -4 and -6 satisfy these conditions: -4 + (-6) = -10 and (-4) * (-6) = 24.
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Use these numbers to create the binomials: (x - 4)(x - 6)
Therefore, the factored form of x² - 10x + 24 is (x - 4)(x - 6).
Conclusion
Understanding how to expand and factor expressions like (x-4)(x-6) is essential in algebra. It allows you to manipulate expressions, solve equations, and simplify complex problems.