- Evaluate the signs of the sum of roots (\(-b/a\)) and product of roots (\(c/a\)) for each function.
- For negative roots, the sum must be negative and the product must be positive.
- Function D, \(x^2 + 9x + 18\), has a sum of \(-9\) and a product of \(18\), meeting the requirements for negative real roots.
- The roots of \(x^2 + 9x + 18 = 0\) are \(-3\) and \(-6\), confirming the answer is \(\boxed{D}\).
Explanation
- Analysis of Conditions for Negative Roots
To determine which of the given quadratic functions has two negative roots, we use the properties of the roots of a quadratic equation \(ax^2 + bx + c = 0\). According to Vieta's formulas, for roots \(r_1\) and \(r_2\):
- The sum of the roots is \(r_1 + r_2 = -\frac{b}{a}\).
- The product of the roots is \(r_1 \cdot r_2 = \frac{c}{a}\).
For both roots to be negative, the following conditions must hold:
- The discriminant \(D = b^2 - 4ac \geq 0\) (to ensure real roots).
- The sum of the roots must be negative (\(-\frac{b}{a} < 0\)).
- The product of the roots must be positive (\(\frac{c}{a} > 0\)).
- Evaluation of Each Function
We evaluate each function based on the identified criteria:
A. \(f(x) = x^2 - 5x - 14\)
- Here, \(a = 1, b = -5, c = -14\).
- Sum of roots: \(-\frac{b}{a} = -\frac{-5}{1} = 5\). Since the sum is positive, the roots cannot both be negative.
B. \(f(x) = x^2 - 10x + 16\)
- Here, \(a = 1, b = -10, c = 16\).
- Sum of roots: \(-\frac{b}{a} = -\frac{-10}{1} = 10\). Since the sum is positive, the roots cannot both be negative.
C. \(f(x) = -x^2 - 8x + 12\)
- Here, \(a = -1, b = -8, c = 12\).
- Sum of roots: \(-\frac{b}{a} = -\frac{-8}{-1} = -8\). This is negative, which is promising.
- Product of roots: \(\frac{c}{a} = \frac{12}{-1} = -12\). Since the product is negative, the roots have opposite signs and cannot both be negative.
D. \(f(x) = x^2 + 9x + 18\)
- Here, \(a = 1, b = 9, c = 18\).
- Sum of roots: \(-\frac{b}{a} = -\frac{9}{1} = -9\). This is negative.
- Product of roots: \(\frac{c}{a} = \frac{18}{1} = 18\). This is positive.
- Discriminant: \(D = b^2 - 4ac = 9^2 - 4(1)(18) = 81 - 72 = 9\). Since \(D > 0\), the roots are real and distinct.
Since all conditions are satisfied for function D, it has two negative roots. Specifically, the roots are \(x = -3\) and \(x = -6\).
- Conclusion
Based on the analysis, function D is the only one that satisfies all conditions for having two negative roots. Therefore, the correct choice is D. \(\boxed{D}\)
Examples
Identifying the signs of roots is useful in physics and engineering, for example, when analyzing the stability of a system described by a characteristic equation. If a system's roots (which often represent decay rates or frequencies) are all negative, the system is typically stable and will return to equilibrium over time. By checking if the roots of the quadratic equation \(x^2 + 9x + 18 = 0\) are negative, we can confirm the stability of a simple model. The roots calculated, \(-3\) and \(-6\), indicate two distinct negative components, confirming the stability.