Find the zeroes of the quadratic polynomial 6x² – 3 – 7x
and verify the relationship between the
zeroes and the coefficients of the polynomial.
please help me
Answers
Solution
Given :-
- Polynomial equation, 6x² - 7x - 3 = 0.
Find :-
- its roots .
- Verification between roots and coefficient
Explanation
Using Formula
★Sum of roots = -(coefficient of x )/(coefficient of x²)
★ product of roots = (constant part)/(coefficient of x²)
So, Now
Let
- Roots be here, p & q
==> p + q = -(-7)/6
==> p + q = 7/6________________(1)
Again,
==> p.q = (-3)/6
==> p.q = -1/2 _________________(2)
By, equ(2)
==> p = -1/2q____________________(3)
Keep in equ(1)
==> -1/2q + q = 7/6
==> -1 + 2q² = 14q/6
==> 2q² - 1 = 7q/3
==> 6q² -7q - 3 = 0
==> 6q² - 9q + 2q - 3 = 0
==> 3q(2q - 3)+1(2q - 3) = 0
==> (3q + 1)(2q - 3) = 0
==> (3q + 1) = 0 Or, (2q - 3) = 0
==> q = -1/3 Or, q = 3/2
keep value p q in equ(3)
When,
- q = -1/3
==> p = -1/(2×-1/3)
==> p = 3/2
When,
- q = 3/2
==> p = -1/(2×3/2)
==> p = -1/3
Hence
- Roots of equation, be 3/2 , -1/3 Or, -1/3 , 3/2
____________________
Now, Find relationship between roots and coefficient
For, this,
==> p + q = 7/6
keep value of p & q
When, p = 3/2 , q = -1/3
==> 3/2 - 1/3 = 7/6
==> (9 - 2)/6 = 7/6
==> 7/6 = 7/6
L.H.S. = R.H.S.
Again,
==> p.q = -1/2
keep value of p & q
When, p = 3/2 , q = -1/3
==> 3/2 × -1/3 = -1/2
==> -1/2 = -1/2
L.H.S. = R.H.S.
That's relationship between roots and coefficient.