Find the Zeros of f(x) = 6x² − 7x − 3 and Verify the Relationship Between Zeros and Coefficients

Video Explanation

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Solution

Given polynomial:

f(x) = 6x² − 3 − 7x

Rearranging in standard form:

f(x) = 6x² − 7x − 3

Step 1: Find the Zeros of the Polynomial

6x² − 7x − 3 = 0

Split the middle term:

6x² − 9x + 2x − 3 = 0

Grouping the terms:

3x(2x − 3) + 1(2x − 3) = 0

(3x + 1)(2x − 3) = 0

∴ 3x + 1 = 0   or   2x − 3 = 0

∴ x = −1/3   or   x = 3/2

Zeros of the polynomial are −1/3 and 3/2.

Step 2: Identify Coefficients

Comparing f(x) = 6x² − 7x − 3 with ax² + bx + c:

a = 6,   b = −7,   c = −3

Step 3: Verify the Relationship

Let α = 3/2 and β = −1/3

Sum of zeros:

α + β = 3/2 + (−1/3)

= (9 − 2)/6 = 7/6

−b/a = −(−7)/6 = 7/6

✔ Sum of zeros = −b/a

Product of zeros:

αβ = (3/2)(−1/3) = −1/2

c/a = −3/6 = −1/2

✔ Product of zeros = c/a

Final Answer

Zeros of the polynomial are 3/2 and −1/3.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, for the quadratic polynomial f(x) = 6x² − 7x − 3, the sum and product of zeros satisfy the standard relationships with its coefficients.