Advertisements
Advertisements
Question
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`-1/4 ,1/4`
Advertisements
Solution
Given: α + β = `-1/4`, αβ = `1/4`
Since ax2 + bx + c = k[x2 - (α + β)x + αβ]
Or `(ax^2 + bx + c)/k = x^2 - (-1/4x) + 1/4)`
Or `(ax^2 + bx + c)/k = (4x^2 + 4x + 1)/4`
Here k is a constant term, by comparing k = 4
Hence, ax2 + bx + c = `4x^2 + 4x + 1`
The quadratic polynomial is `4x^2 + 4x + 1`.
APPEARS IN
RELATED QUESTIONS
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`g(x)=a(x^2+1)-x(a^2+1)`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha-1/beta`
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − px + q, prove that `alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2)/q+2`
If the sum of the zeros of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k.
If one zero of the quadratic polynomial f(x) = 4x2 − 8kx − 9 is negative of the other, find the value of k.
If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.
If two zeros x3 + x2 − 5x − 5 are \[\sqrt{5}\ \text{and} - \sqrt{5}\], then its third zero is
The product of the zeros of x3 + 4x2 + x − 6 is
A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is
If two zeroes of the polynomial x3 + x2 − 9x − 9 are 3 and −3, then its third zero is
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
If the sum of the roots is –p and the product of the roots is `-1/"p"`, then the quadratic polynomial is:
If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the product of the other two zeroes is ______.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2x^2 + (7/2)x + 3/4`
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2s^2 - (1 + 2sqrt(2))s + sqrt(2)`
The zeroes of the quadratic polynomial x2 + 99x + 127 are ______.
If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of the polynomial 2x2 – 5x – 3, then find the values of p and q.
Find the sum and product of the roots of the quadratic equation 2x2 – 9x + 4 = 0.
