Question:
If \(a\) is an integer lying in \([-5,30]\), then the probability that the graph of \(y=x^{2}+2(a+4)x-5a+64\) lies above the x-axis is:
If \(a\) is an integer lying in \([-5,30]\), then the probability that the graph of \(y=x^{2}+2(a+4)x-5a+64\) lies above the x-axis is:
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Remember to always include the endpoints when counting integers inside a closed interval $[p, q]$. The shortcut formula is always $\text{Count} = q - p + 1$. Forgetting to add that extra 1 is a very common slip-up!
Updated On: May 28, 2026
- $\frac{1}{6}$
- $\frac{7}{36}$
- $\frac{2}{9}$
- $\frac{3}{5}$
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The Correct Option is C
Solution and Explanation
Concept:
For an upward-opening quadratic parabola function $y = Ax^2 + Bx + C$ (where $A > 0$) to lie entirely above the horizontal $x$-axis for all real numbers, the equation must have no real roots. This geometric condition requires that the mathematical discriminant must be strictly negative:
$$D = B^2 - 4AC < 0$$
Step 1: Count the total size of the sample space.
We are given that $a$ is an integer chosen from the closed interval $[-5, 30]$. Let us count the total number of integer options available in this range: $$\text{Total Integers } (N) = 30 - (-5) + 1 = 36 \text{ options}$$
Step 2: Set up and solve the discriminant inequality.
For the given quadratic equation $y = x^2 + 2(a+4)x - 5a + 64$, the coefficients are $A = 1$, $B = 2(a+4)$, and $C = -5a + 64$. Calculate the discriminant: $$D = [2(a+4)]^2 - 4(1)(-5a + 64) < 0$$ $$4(a^2 + 8a + 16) + 20a - 256 < 0$$ Divide the entire inequality expression by 4 to simplify the terms: $$(a^2 + 8a + 16) + 5a - 64 < 0$$ $$a^2 + 13a - 48 < 0$$
Step 3: Find the valid range for the parameter $a$.
Factor the quadratic inequality expression: $$(a + 16)(a - 3) < 0$$ Using the standard sign-chart method, the product is negative when $a$ lies strictly between the two root zero points: $$-16 < a < 3 \quad \cdots (1)$$
Step 4: Count the favorable integers matching both constraints.
Now intersect our valid range condition from equation (1) with the problem's given interval constraint $[-5, 30]$: $$\text{Valid range: } a \in [-5, 30] \cap (-16, 3) \quad \Rightarrow \quad a \in [-5, 2]$$ Let us list and count the favorable integers inside this intersection set $[-5, 2]$: $$\text{Favorable integers} = \{-5, -4, -3, -2, -1, 0, 1, 2\}$$ $$\text{Count } (n) = 2 - (-5) + 1 = 8 \text{ integers}$$
Step 5: Calculate the final probability fraction.
$$\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{n}{N} = \frac{8}{36} = \frac{2}{9}$$ This matches option (C) perfectly.
We are given that $a$ is an integer chosen from the closed interval $[-5, 30]$. Let us count the total number of integer options available in this range: $$\text{Total Integers } (N) = 30 - (-5) + 1 = 36 \text{ options}$$
Step 2: Set up and solve the discriminant inequality.
For the given quadratic equation $y = x^2 + 2(a+4)x - 5a + 64$, the coefficients are $A = 1$, $B = 2(a+4)$, and $C = -5a + 64$. Calculate the discriminant: $$D = [2(a+4)]^2 - 4(1)(-5a + 64) < 0$$ $$4(a^2 + 8a + 16) + 20a - 256 < 0$$ Divide the entire inequality expression by 4 to simplify the terms: $$(a^2 + 8a + 16) + 5a - 64 < 0$$ $$a^2 + 13a - 48 < 0$$
Step 3: Find the valid range for the parameter $a$.
Factor the quadratic inequality expression: $$(a + 16)(a - 3) < 0$$ Using the standard sign-chart method, the product is negative when $a$ lies strictly between the two root zero points: $$-16 < a < 3 \quad \cdots (1)$$
Step 4: Count the favorable integers matching both constraints.
Now intersect our valid range condition from equation (1) with the problem's given interval constraint $[-5, 30]$: $$\text{Valid range: } a \in [-5, 30] \cap (-16, 3) \quad \Rightarrow \quad a \in [-5, 2]$$ Let us list and count the favorable integers inside this intersection set $[-5, 2]$: $$\text{Favorable integers} = \{-5, -4, -3, -2, -1, 0, 1, 2\}$$ $$\text{Count } (n) = 2 - (-5) + 1 = 8 \text{ integers}$$
Step 5: Calculate the final probability fraction.
$$\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{n}{N} = \frac{8}{36} = \frac{2}{9}$$ This matches option (C) perfectly.
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