Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and their coefficients. (i) 63 - 2x - x2 (ii) 49x2 - 81
Answers
Step-by-step explanation:
Given :-
(i) 63 - 2x - x²
(ii) 49x² - 81
To find :-
Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and their coefficients.
Solution :-
i)
Given Quadratic Polynomial is 63-2x-x²
Let P(x) = 63-2x-x²
On writting this in the standard form
P(x) = -x²-2x+63
Finding the zeroes :-
To get the zeroes we write P(x) = 0
=> -x²-2x+63 = 0
=> -(x²+2x-63) = 0
=> x²+2x-63 = 0
=> x²+9x-7x-63 = 0
=> x(x+9)-7(x+9) = 0
=> (x+9)(x-7) = 0
=> x+9 = 0 or x-7 = 0
=> x = -9 or x = 7
The zeroes are -9 and 7
Finding the relationship between the zeroes and the coefficients:-
P(x) = -x²-2x+63
On Comparing this with the standard quadratic Polynomial ax²+bx+c
a = -1
b= -2
c = 63
The zeroes are -9 and 7
Let α = -9 and β = 7
Sum of the zeroes = α +β
=> -9+7
=> -2
=> α + β = -2
Sum of the zeroes = -b/a
=> -(-2)/-1
=> 2/-1
=> -2
Sum of the zeroes =α+ β = -b/a
Product of the zeroes = α β
=> α β = (-9)(7)
=> α β = -63
Product of the zeroes = c/a
=> 63/-1
=> -63
Product of the zeroes = α β = c/a
________________________________
ii)
Given Quadratic Polynomial is 49x²-81
Let P(x) = 49x²-81
Finding the zeroes :-
To get the zeroes we write P(x) = 0
=> 49x²-81= 0
=>7²x²-9²= 0
=>(7x)²-9²= 0
=> (7x+9)(7x-9) = 0
Since (a+b)(a-b) = a²-b²
Where , a = 7x and b = 9
=> 7x+9 = 0 or 7x-9 = 0
=> 7x = -9 or 7x= 9
=> x = -9/7 or x = 9/7
The zeroes are -9/7 and 9/7
Finding the relationship between the zeroes and the coefficients:-
P(x) =49x²-81
On Comparing this with the standard quadratic Polynomial ax²+bx+c
a = 49
b= 0
since there is no term with coefficient of x
c =-81
The zeroes are -9/7 and 9/7
Let α = -9/7 and β = 9/7
Sum of the zeroes = α +β
=> (-9/7)+(9/7)
=> (-9+9)/7
=> 0/7
=> 0
=> α + β = 0
Sum of the zeroes = -b/a
=> 0/49
=>0
Sum of the zeroes =α+ β = -b/a
Product of the zeroes = α β
=> α β = (-9/7)(9/7)
=> α β = (-9×9)/(7×7)
=>α β=-81/49
Product of the zeroes = c/a
=> -81/49
Product of the zeroes = α β = c/a
Answer:-
i) The zeroes are -9 and 7
ii)The zeroes are -9/7 and 9/7
Verified the given relations in the given problems.
Used formulae:-
- The standard quadratic polynomial is ax²+bx+c
- Sum of the zeroes =α+ β = -b/a
- Product of the zeroes = α β = c/a
- (a+b)(a-b) = a²-b²
- To get the zeroes of P(x) we can write it as P(x) = 0.