The probability of selecting integers a ∈ [-5, 30] such that x² + 2(a + 4)x - 5a + 64 0, for all x

Q: Why is it $D < 0$ and not $D \le 0$?

That is a fantastic observation! If $D = 0$, the parabola touches the x-axis at exactly one point, meaning $f(x) = 0$ for that specific $x$. However, the problem demands that $f(x) > 0$ for *all* real $x$. Since zero is not strictly greater than zero, we must exclude the case where the parabola touches the axis. Thus, we strictly require $D < 0$.

Q: Why do we use $U - L + 1$ to count integers?

This is a classic 'fencepost error' that catches many students. Think of it this way: if you want to count the integers from 1 to 3, you have 1, 2, and 3, which is 3 integers. If you just did $3 - 1$, you would get 2. You must add the 1 to account for the inclusion of both endpoints. It is a fundamental rule for counting discrete values in a closed interval.

Q: Why do we intersect the intervals?

The problem imposes two distinct constraints. First, the algebraic condition for the quadratic to be positive, which gives us a range for $a$. Second, the problem restricts our choice of $a$ to the set $[-5, 30]$. Since $a$ must satisfy *both* conditions simultaneously, we must find the overlap—the intersection—of these two sets.

Question

The probability of selecting integers $a \in [-5, 30]$ such that $x^2 + 2(a + 4)x - 5a + 64 > 0$, for all $x \in \mathbb{R}$, is:

Options:

(A) $7/36$

(B) $2/9$

(C) $1/6$

(D) $1/4$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The problem asks us to find the probability of selecting an integer $a$ such that the quadratic expression $f(x) = x^2 + 2(a + 4)x - 5a + 64$ is strictly greater than zero for all real values of $x$.

Visually, for a quadratic expression to be always positive, its graph must hover entirely above the $x$-axis. It cannot touch or cross the axis, meaning it must have no real roots.

### The Discriminant: The Gatekeeper of Roots

To ensure the parabola stays above the $x$-axis, we must satisfy the condition that the discriminant $D = B^2 - 4AC$ is strictly less than zero. Since the leading coefficient $A = 1$ is positive, the parabola opens upwards, and $D < 0$ guarantees that the entire curve remains above the $x$-axis.

We identify the coefficients as follows:
$A = 1$
$B = 2(a + 4)$
$C = -5a + 64$

### The Algebraic Dance

Substituting these coefficients into the discriminant formula, we obtain:

$$D = [2(a + 4)]^2 - 4(1)(-5a + 64) < 0$$

Dividing the entire inequality by $4$ simplifies the expression to:

$$(a + 4)^2 - (-5a + 64) < 0$$

Expanding the terms, we get:

$$a^2 + 8a + 16 + 5a - 64 < 0$$

Combining like terms leads to the quadratic inequality:

$$a^2 + 13a - 48 < 0$$

### Solving the Inequality

We factorize the quadratic expression by finding two numbers that multiply to $-48$ and add to $13$. These numbers are $16$ and $-3$.

The inequality becomes:

$$(a + 16)(a - 3) < 0$$

This implies that the value of $a$ must lie within the interval:

$$a \in (-16, 3)$$

### The Final Constraint and Probability

We are given that $a$ is selected from the interval $[-5, 30]$. We must find the intersection of our calculated range $(-16, 3)$ and the given constraint $[-5, 30]$, which results in the interval $[-5, 3)$.

The integers contained in this interval are $\{-5, -4, -3, -2, -1, 0, 1, 2\}$. There are exactly $8$ such integers.

The total number of integers in the range $[-5, 30]$ is calculated as:

$$30 - (-5) + 1 = 36$$

The probability $P$ is the ratio of favorable outcomes to total outcomes:

$$P = \frac{8}{36} = \frac{2}{9}$$

The final probability is **$\frac{2}{9}$**.

Animated Solution for Mathematics - Quadratic Equations: The probability of selecting integers $a \in [- 5, 30]$ such that $x^{2} + 2 (a + 4) x - 5 a + 64 > 0$ , for all $x \in R$ , is:

Select Answer:

Visualized Solution

Visualizing the Condition $f (x) > 0$

Setting up the Discriminant $D < 0$

Raw Setup of Discriminant

Expanding the Inequality

Simplifying the Quadratic in $a$

Factorizing the Expression

Finding the Range of $a$

The Given Constraint

Identifying Favorable Integers

Counting Favorable Outcomes

Calculating Total Possible Integers

Final Probability Calculation

The Sigma Insight: Maximum and Minimum Values of Quadratic Expressions

Analyzing the Setup

The Discriminant

The Algebraic Dance

Solving the Inequality

The Final Constraint and Probability

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability of selecting integers a∈[−5,30] such that x2+2(a+4)x−5a+64>0, for all x∈R, is: