Identifying the Correct Quadratic Equation
To determine which of the given quadratic equations has a root of 2, we can substitute 2 into both equations and check if it satisfies them.
Given Quadratic Equations:
- (a) x² + 4x - 1 = 0
- (b) x² - 4x - 1 = 0
Substituting the Root:
- For equation (a):
- Substitute x = 2:
- 2² + 4(2) - 1 = 4 + 8 - 1 = 11 (not equal to 0)
- For equation (b):
- Substitute x = 2:
- 2² - 4(2) - 1 = 4 - 8 - 1 = -5 (not equal to 0)
Conclusion:
Neither of the provided equations has 2 as a root, as neither evaluates to zero when 2 is substituted.
Finding a Correct Quadratic Equation:
To create a quadratic equation with 2 as a root, we can use the fact that if x = 2 is a root, then (x - 2) is a factor of the equation.
- A simple form of the equation can be constructed:
- f(x) = (x - 2)(x - r) where r is another root.
For instance, if we choose r = 3, we get:
- f(x) = (x - 2)(x - 3) = x² - 5x + 6.
This indicates that we can create various quadratic equations with 2 as a root by selecting different values for r.
Final Note:
Ensure to verify any quadratic equation by substituting the proposed roots to confirm their validity.