Relationship Between Roots and Coefficients
If the two roots of ax² + bx + c = 0 are α, β:
α + β = -b/a、αβ = c/a
Example ①
If the two roots of x² - 5x + 3 = 0 are α, β, find α + β and αβ.
Answer: α + β = 5、αβ = 3
If the two roots of x² - 5x + 3 = 0 are α, β, find α + β and αβ.
Answer: α + β = 5、αβ = 3
Example ②
Construct a quadratic equation whose two roots are 3 and -2.
Answer: The sum is 1 and the product is -6, so x² - x - 6 = 0
Construct a quadratic equation whose two roots are 3 and -2.
Answer: The sum is 1 and the product is -6, so x² - x - 6 = 0
Key Point
- Roots and coefficients: α + β = -b/a、αβ = c/a
- Conversely, an equation from two roots: x² - (α+β)x + αβ = 0
- Transformations such as α² + β² = (α+β)² - 2αβ also frequently appear
Practice Problems
- For the two roots α, β of x² + 4x - 7 = 0, find α² + β².
Answers and Explanations
- Answer: 30. α+β=-4, αβ=-7. α²+β²=(α+β)²-2αβ=16+14=30.