Find the Zeros of q(x) = √3x² + 10x + 7√3 and Verify the Relationship Between Zeros and Coefficients

Video Explanation

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Solution

Given polynomial:

q(x) = √3x² + 10x + 7√3

Step 1: Find the Zeros of the Polynomial

√3x² + 10x + 7√3 = 0

Divide the whole equation by √3:

x² + (10/√3)x + 7 = 0

Split the middle term:

x² + (7/√3)x + (3/√3)x + 7 = 0

Grouping the terms:

x(x + 7/√3) + 1(x + 7/√3) = 0

(x + 7/√3)(x + 1) = 0

∴ x + 7/√3 = 0   or   x + 1 = 0

∴ x = −7/√3   or   x = −1

Zeros of the polynomial are −7/√3 and −1.

Step 2: Identify Coefficients

Comparing q(x) = √3x² + 10x + 7√3 with ax² + bx + c:

a = √3,   b = 10,   c = 7√3

Step 3: Verify the Relationship

Let α = −7/√3 and β = −1

Sum of zeros:

α + β = −7/√3 − 1

= −(7 + √3)/√3

−b/a = −10/√3

✔ Sum of zeros = −b/a

Product of zeros:

αβ = (−7/√3)(−1) = 7/√3

c/a = (7√3)/(√3) = 7/√3

✔ Product of zeros = c/a

Final Answer

Zeros of the polynomial are −7/√3 and −1.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, for the quadratic polynomial q(x) = √3x² + 10x + 7√3, the sum and product of zeros satisfy the standard relationships with its coefficients.