Question:

Step 1: Ensure the equation is quadratic For the given equation to be quadratic, the coefficient of $x^2$ must be non-zero. Thus, $4p^2 - p - 5 \neq 0$. To find the values of $p$ for which this expression is zero, we solve the quadratic equation $4p^2 - p - 5 = 0$: $$p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-5)}}{2(4)}$$ $$p = \frac{1 \pm \sqrt{1 + 80}}{8}$$ $$p = \frac{1 \pm \sqrt{81}}{8}$$ $$p = \frac{1 \pm 9}{8}$$ This yields two values for $p$: $$p_1 = \frac{1 + 9}{8} = \frac{10}{8} = \frac{5}{4}$$ $$p_2 = \frac{1 - 9}{8} = \frac{-8}{8} = -1$$ Therefore, for the equation to be quadratic, $p \neq \frac{5}{4}$ and $p \neq -1$. Step 2: Apply the condition for roots on either side of unity Let $f(x) = (4p^2 - p - 5)x^2 - (2p - 1)x + 3p$. For the roots of a quadratic equation $ax^2 + bx + c = 0$ to lie on either side of a number $k$, the condition is $a \cdot f(k) < 0$. In this case, $a = 4p^2 - p - 5$ and $k = 1$. First, calculate $f(1)$: $$f(1) = (4p^2 - p - 5)(1)^2 - (2p - 1)(1) + 3p$$ $$f(1) = 4p^2 - p - 5 - 2p + 1 + 3p$$ $$f(1) = 4p^2 - 4$$ $$f(1) = 4(p^2 - 1)$$ $$f(1) = 4(p - 1)(p + 1)$$ Step 3: Set up the inequality The condition for roots to lie on either side of unity is $a \cdot f(1) < 0$. Substitute the expressions for $a$ and $f(1)$: $$(4p^2 - p - 5) \cdot 4(p - 1)(p + 1) < 0$$ Factor the quadratic term $4p^2 - p - 5$: $$4p^2 - p - 5 = (4p - 5)(p + 1)$$ Now substitute this back into the inequality: $$(4p - 5)(p + 1) \cdot 4(p - 1)(p + 1) < 0$$ $$4(p - 1)(p + 1)^2(4p - 5) < 0$$ Step 4: Analyze the sign We need to solve the inequality $4(p - 1)(p + 1)^2(4p - 5) < 0$. Since $4 > 0$, we can divide by 4 without changing the inequality direction: $$(p - 1)(p + 1)^2(4p - 5) < 0$$ The term $(p + 1)^2$ is always non-negative. For the entire expression to be strictly less than zero, $(p + 1)^2$ must be strictly positive, which means $p + 1 \neq 0$, so $p \neq -1$. This condition is consistent with Step 1. If $(p + 1)^2 > 0$, we can divide by it, and the inequality simplifies to: $$(p - 1)(4p - 5) < 0$$ To find the values of $p$ that satisfy this inequality, we identify the roots of $(p - 1)(4p - 5) = 0$, which are $p = 1$ and $p = \frac{5}{4}$. By testing values in the intervals defined by these roots, we find that $(p - 1)(4p - 5) < 0$ when $p$ is between the roots: $$1 < p < \frac{5}{4}$$ This interval $1 < p < \frac{5}{4}$ already satisfies the conditions $p \neq -1$ and $p \neq \frac{5}{4}$. Step 5: Find integral values We need to find the number of integral values of $p$ in the interval $1 < p < \frac{5}{4}$. Converting the fraction to a decimal, we have $1 < p < 1.25$. There are no integers in the open interval $(1, 1.25)$. Therefore, the number of integral values of $p$ is 0.