Question

Find the value of p, for which one zero of the quadratic polynomial \(px^2 - 14x + 8\) is 6 times the other.

• A. 3
• B. 2
• C. 4
• D. 1

Hint:

For any quadratic polynomial where one root is \(n\) times the other:
The relation between coefficients is:
\[ (n + 1)^2 ac = n b^2 \]
Here, \(n = 6\), \(a = p\), \(b = -14\), and \(c = 8\):
\[ (6 + 1)^2 \times p \times 8 = 6 \times (-14)^2 \]
\[ 49 \times 8 \times p = 6 \times 196 \]
Since \(196 = 49 \times 4\):
\[ 8p = 6 \times 4 \implies 8p = 24 \implies p = 3 \]
This general formula is extremely fast and useful for saving time!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a quadratic polynomial \(f(x) = px^2 - 14x + 8\). We are told that one of its zeroes is exactly 6 times the other zero. We need to determine the value of the quadratic coefficient \(p\).

Step 2: Key Formula or Approach:
1. Let the two zeroes of the polynomial be \(\alpha\) and \(6\alpha\).
2. Use Vieta's formulas relating the sum and product of zeroes to the coefficients of \(ax^2 + bx + c\):
\[ \text{Sum of zeroes } = -\frac{b}{a} \]
\[ \text{Product of zeroes } = \frac{c}{a} \]

Step 3: Detailed Explanation:
1. Identify the coefficients of the polynomial \(px^2 - 14x + 8\):
\[ a = p, \quad b = -14, \quad c = 8 \]
2. Write the equation for the sum of the zeroes:
\[ \alpha + 6\alpha = -\frac{-14}{p} \]
\[ 7\alpha = \frac{14}{p} \]
Divide both sides by 7:
\[ \alpha = \frac{2}{p} \quad \text{--- (Equation 1)} \]
3. Write the equation for the product of the zeroes:
\[ \alpha \times 6\alpha = \frac{8}{p} \]
\[ 6\alpha^2 = \frac{8}{p} \quad \text{--- (Equation 2)} \]
4. Substitute the value of \(\alpha\) from Equation 1 into Equation 2:
\[ 6\left( \frac{2}{p} \right)^2 = \frac{8}{p} \]
\[ 6\left( \frac{4}{p^2} \right) = \frac{8}{p} \]
\[ \frac{24}{p^2} = \frac{8}{p} \]
5. Since \(p \neq 0\) (as it is the leading coefficient of a quadratic polynomial), multiply both sides of the equation by \(p^2\):
\[ 24 = 8p \]
\[ p = \frac{24}{8} = 3 \]
This gives the value of \(p\) as 3.

Step 4: Final Answer:
The value of \(p\) is 3, which corresponds to option (A).