(x-b)-formula.jpeg "(x+a)(x-b) Formula")
Understanding the (x+a)(x-b) Formula
The formula (x+a)(x-b) represents the product of two binomials, where x is a variable, and a and b are constants. This formula is a fundamental concept in algebra and finds its application in simplifying expressions, solving equations, and factoring polynomials.
Expanding the Formula
To expand the formula, we use the distributive property of multiplication, which states that multiplying a sum by a number is the same as multiplying each term of the sum by the number:
(x + a)(x - b) = x(x - b) + a(x - b)
Then, we distribute again:
= x² - bx + ax - ab
Finally, we combine the terms with x:
= x² + (a - b)x - ab
Therefore, the expanded form of (x + a)(x - b) is x² + (a - b)x - ab.
Applications
The (x+a)(x-b) formula has various applications in algebra, including:
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Factoring quadratic expressions: If we have a quadratic expression in the form x² + (a - b)x - ab, we can easily factor it into (x + a)(x - b).
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Solving quadratic equations: By factoring a quadratic equation using the (x + a)(x - b) formula, we can find its solutions.
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Simplifying algebraic expressions: This formula can help simplify complex algebraic expressions by expanding or factoring them.
Example
Let's say we want to factor the expression x² + 5x - 6. We can use the (x+a)(x-b) formula to do this:
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Identify a and b: In this case, a = 6 and b = 1 because 6 - 1 = 5 and 6 * 1 = 6.
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Substitute a and b into the formula: This gives us (x + 6)(x - 1).
Therefore, the factored form of x² + 5x - 6 is (x + 6)(x - 1).
Conclusion
The (x+a)(x-b) formula is a valuable tool in algebra for expanding, factoring, and simplifying expressions. It provides a straightforward method for working with quadratic equations and helps solve various mathematical problems efficiently. Mastering this formula is crucial for a strong foundation in algebra.