Advertisements
Advertisements
Question
Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?
Advertisements
Solution
A Quadratic Equation will have equal roots if it satisfies the condition:
b2 – 4ac = 0
Given equation is x2 + kx + k = 0
a = 1, b = k, x = k
Substituting in the equation we get,
k2 – 4(1)(k) = 0
k2 – 4k = 0
k(k – 4) = 0
k = 0, k = 4
But in the question, it is given that k is greater than 1.
Hence the value of k is 4 if the equation has common roots.
Hence if the value of k = 4, then the equation (x2 + kx + k) will have equal roots.
APPEARS IN
RELATED QUESTIONS
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
4, 1
Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)` and `(2 - sqrt3)`
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
`p(x) = x^2 + 2sqrt2x + 6`
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`f(x)=x^2-(sqrt3+1)x+sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha+1/beta-2alphabeta`
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2, find a quadratic polynomial whose zeroes are `1/(2alpha+beta)+1/(2beta+alpha)`
Find the quadratic polynomial, the sum of whose zeroes is `(5/2)` and their product is 1. Hence, find the zeros of the polynomial.
If (x+a) is a factor of the polynomial `2x^2 + 2ax + 5x + 10`, find the value of a.
Verify that 5, –2 and `1/3` are the zeroes of the cubic polynomial p(x) = (3x3 – 10x2 – 27x + 10) and verify the relation between its zeros and coefficients.
Find a cubic polynomial whose zeroes are `1/2`, 1 and –3.
Find the quotient and the remainder when f(x) = x4 – 5x + 6 is divided by g(x) = 2 – x2.
If α, β are the zeroes of the polynomial f(x) = 5x2 – 7x + 1 then `1/α + 1/β = ?`
If α, β are the zeroes of the polynomial f(x) = x2 + x – 2, then `(α/β - α/β)`.
Define a polynomial with real coefficients.
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`-2sqrt(3), -9`
The zeroes of the quadratic polynomial x2 + 99x + 127 are ______.
If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it ______.
If α, β are the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then `(1/α + 1/β)` is equal to ______.
