Find the zeroes of the following quadratic polynomial and verify the

1.	Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.6x2 – 3 – 7x
Answer» Given polynomial is 6x²– 3 – 7x We have, 6x² – 3 – 7x = 6x² – 7x – 3 = 6x² – 9x + 2x – 3 = 3x(2x – 3) + 1(2x – 3) = (2x – 3) (3x + 1) The value of 6x² – 3 – 7x is zero, when the value of (3x +1) (2x – 3) is 0 i.e., when 3x + 1 = 0 and 2x – 3 = 0 3x = -1 and 2x = 3 x = -1/3 and x = 3/2 ∴ The zeroes of 6x² – 3 – 7x = -1/3 and 3/2 ∴ Sum of the zeroes = 1/3 + 3/2 = 7/6. = \(-\frac{Coefficient\,of\,x}{Coefficient\,of\,x^2}=\frac{-(-7)}{6}=\frac{7}{6}\) And product of the zeroes = (-1/3) x (3/2) = -1/2 = \(\frac{Constant\,term}{Coefficient\,of\,x^2}= \frac{-3}{6}=\frac{-1}{2}\)

Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.6x2 – 3 – 7x

Answer»

Given polynomial is 6x²– 3 – 7x

We have, 6x² – 3 – 7x = 6x² – 7x – 3

= 6x² – 9x + 2x – 3

= 3x(2x – 3) + 1(2x – 3)

= (2x – 3) (3x + 1)

The value of 6x² – 3 – 7x is zero, when the value of (3x +1) (2x – 3) is 0

i.e., when 3x + 1 = 0 and 2x – 3 = 0

3x = -1 and 2x = 3

x = -1/3 and x = 3/2

∴ The zeroes of 6x² – 3 – 7x = -1/3 and 3/2

∴ Sum of the zeroes = 1/3 + 3/2 = 7/6.

= \(-\frac{Coefficient\,of\,x}{Coefficient\,of\,x^2}=\frac{-(-7)}{6}=\frac{7}{6}\)

And product of the zeroes = (-1/3) x (3/2) = -1/2

= \(\frac{Constant\,term}{Coefficient\,of\,x^2}= \frac{-3}{6}=\frac{-1}{2}\)