The equation a * x ^ 2 + bx + c = 0 , where a, b, c are real numbers, has one root greater than 2 and the other root less than zero. Which of the following is necessarily true? (1) a(a + b + c) > 0 (2) a(a + b + c) < 0 (3) a + b + c > 0 (4) a + b + c < 0
The Quadratic Equation and Root Analysis
Answer: The quadratic equation given is ax^2 + bx + c = 0, where a, b, and c are real numbers. It is stated that one root is greater than 2 and the other is less than 0. We need to determine which of the given options is necessarily true.
Understanding the Roots of the Quadratic Equation
A quadratic equation of the form ax^2 + bx + c = 0 can be solved using the quadratic formula:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The roots of the equation depend on the discriminant, D = b^2 - 4ac, and the quotient of the division by 2a.
Given that one root is greater than 2 and the other is less than zero, we denote these roots by r_1 and r_2 such that r_1 > 2 and r_2 < 0.
Properties of the Roots and Vieta’s Formulas
Using Vieta’s formulas, we know:
- The sum of the roots is given by r_1 + r_2 = -\frac{b}{a}.
- The product of the roots is given by r_1 \cdot r_2 = \frac{c}{a}.
Constraints from Root Conditions
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From the root condition r_1 \cdot r_2 = \frac{c}{a} and r_1 > 2, r_2 < 0:
- The product r_1 \cdot r_2 < 0 as one root is positive and the other is negative, implying \frac{c}{a} < 0.
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From the sum r_1 + r_2 = -\frac{b}{a}:
- Given r_1 > 2 and r_2 < 0, r_1 + r_2 can be either positive or negative. However, it provides no direct information about the sign of \frac{-b}{a}.
Analyzing the Expression a(a + b + c)
Next, let us consider the expression a(a + b + c) and its relation to the root conditions.
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Expand:
a(a + b + c) = a^2 + ab + ac -
Connect with Vieta:
- From Vieta: r_1 + r_2 = -b/a \Rightarrow b = -a(r_1 + r_2)
- From Vieta: r_1 \cdot r_2 = c/a \Rightarrow c = a(r_1 \cdot r_2)
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Substitute these into a + b + c:
- a + b + c = a - a(r_1 + r_2) + a(r_1 \cdot r_2).
Exploring Options
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Option 1: a(a + b + c) > 0
By substituting and expanding, we have:
- a(a + b + c) = a^2 - a^2(r_1 + r_2) + a^2(r_1 \cdot r_2).
Since a^2 > 0, for this product to be positive, it must align with root conditions.
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Option 2: a(a + b + c) < 0
Similarly, for a^2 - a^2(r_1 + r_2) + a^2(r_1 \cdot r_2) < 0 to be true, the signs and relative sizes of expressions derived from Vieta’s formulas must dictate this negativity.
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Option 3: a + b + c > 0
Directly checking:
- Positive: Could align under certain conditions of roots and coefficients which maintain a + b + c > 0 directly.
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Option 4: a + b + c < 0
Again, checking accordingly if the arrangement of roots and parameters allows this negativity naturally with the problem’s conditions.
Conclusion
Given the problem’s constraints, we recognize the negativity from roots configuration directly affects r_1 \cdot r_2 < 0. Further analysis finds option alignments conditional upon a, b, c values under r_1 > 2 and r_2 < 0. However, the collective inference identifies that option 2, a(a + b + c) < 0, best fits a broader necessary criterion with constraints within comparative exercises, given inherent properties of negative-dispositional roots influencing discriminant determinant influences and derived conditions.
Thus, option 2 mirrors most consistent inherent necessity.
[To further understand or explore radical expressions and discriminantal exploration under different bounds, consider highly interactive and systematic exercises from algebraic mathematic literature resources.]