(ii) 8x² – 14x – 15. Find the zeroes of this quadratic polynomial and verify the relationship between the zeroes and coefficients.
The Correct Answer and Explanation is:
Correct Answer:
Given quadratic polynomial:
8x² – 14x – 15
Step 1: Find the zeroes using factorization
To factor 8x² – 14x – 15, first find two numbers whose product is:
(8)(-15) = -120
and whose sum is:
-14
The two numbers are 6 and -20 because:
6 × (-20) = -120
6 + (-20) = -14
Now, split the middle term:
8x² + 6x – 20x – 15
Group terms:
(8x² + 6x) – (20x + 15)
Factor each group:
2x(4x + 3) -5(4x + 3)
Factor out the common binomial:
(4x + 3)(2x – 5)
Set each factor equal to zero:
- 4x + 3 = 0 → x = -3/4
- 2x – 5 = 0 → x = 5/2
So, the zeroes of the polynomial are x = -3/4 and x = 5/2
Step 2: Verify relationship between zeroes and coefficients
For a quadratic polynomial in the form ax² + bx + c, the relationships are:
- Sum of zeroes = -b/a
- Product of zeroes = c/a
From the polynomial 8x² – 14x – 15:
a = 8, b = -14, c = -15
Sum of zeroes:
(-3/4) + (5/2) = (-3 + 10)/4 = 7/4
-b/a = -(-14)/8 = 14/8 = 7/4 ✔️
Product of zeroes:
(-3/4)(5/2) = -15/8
c/a = -15/8 ✔️
Conclusion:
The zeroes of the quadratic polynomial 8x² – 14x – 15 are -3/4 and 5/2. These zeroes satisfy the relationships between the sum and product of zeroes and the coefficients, confirming the correctness of the factorization and computation.
