To find the factors of the quadratic expression x² + 4x – 210, we can follow a systematic approach. Here’s how you can do it:
- Identify the Coefficients: The expression is in the standard form of ax² + bx + c, where:
- a = 1 (the coefficient of x²)
- b = 4 (the coefficient of x)
- c = -210 (the constant term)
- Use the Factoring Method: We are looking for two numbers that multiply to ac (which is 1 * -210 = -210) and add up to b (which is 4).
- Find the Pair: After checking the pairs of factors of -210, we can determine that the numbers 15 and -14 meet these criteria:
- 15 * -14 = -210
- 15 + (-14) = 1
- Rewrite the Expression: We can express the middle term (4x) using our pair of numbers:
- x² + 15x – 14x – 210
- Factor by Grouping: Next, we group the terms:
- (x² + 15x) + (-14x – 210)
- Factoring out the common terms from each group gives us:
- x(x + 15) – 14(x + 15)
- Final Factorization: Now, we factor out the common binomial factor:
- (x + 15)(x – 14)
Thus, the factors of the expression x² + 4x – 210 are (x + 15) and (x – 14).
To verify, you can expand these factors:
- (x + 15)(x – 14) = x² – 14x + 15x – 210 = x² + 4x – 210
This confirms that the factorization is correct. Understanding this factorization technique can help with solving quadratic equations and optimizing polynomial expressions in the future!