The quadratic equation whose roots are 9 and 11 is:

Question

Examoogle · Accepted Answer

The correct answer is Option A: $x^{2} - 20x + 99 = 0$. Observe that the coefficient of x is the negative sum of the roots (-(9+11) = -20) and the constant term is the product (9*11 = 99). Only option A matches both conditions.

Step-by-Step Solution & Concepts:
Shortcut Trick: Observe that the coefficient of x is the negative sum of the roots (-(9+11) = -20) and the constant term is the product (9*11 = 99). Only option A matches both conditions.

Formula Used: $$x^{2} - (\alpha + \beta)x + (\alpha 	imes \beta) = 0$$

Given: The roots of the quadratic equation are 9 and 11.

Steps: Identify the roots: Let the roots of the quadratic equation be $\alpha = 9$ and $\beta = 11$.. Calculate the sum of roots: Find the sum of the roots: $\alpha + \beta = 9 + 11 = 20$.. Calculate the product of roots: Find the product of the roots: $\alpha 	imes \beta = 9 	imes 11 = 99$.. Form the equation: Substitute the values into the standard quadratic form $x^{2} - (sum)x + (product) = 0$.

Answer: A is correct because substituting the sum 20 and product 99 into the standard quadratic formula yields x^2 - 20x + 99 = 0.

Exam Tip: Understanding Vieta's formulas is crucial for solving problems involving the relationship between coefficients and roots in higher-degree polynomials.

Examoogle · Answer

Option B: $x^{2} + 20x + 100 = 0$

Examoogle · Answer

Option C: $x^{2} - 20x + 98 = 0$

Examoogle · Answer

Option D: $x^{2} + 20x - 99 = 0$

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