Advertisements
Advertisements
Question
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`1/4 , -1`
Advertisements
Solution
Given: α + β = `1/4`, αβ = -1
Since ax2 + bx + c = kx2 - k(α + β)x + kαβ
In comparison,
a = k, b = -k(α + β) and c = kαβ
α + β = `(-b)/a = 1/4` and αβ = `c/a = -1`
⇒ a = 4
⇒ b = -4(α + β)
⇒ c = kαβ = 4(-1)
Hence, on writing as ax2 + bx + c
⇒ 4x2 - 4(α + β)x + 4(αβ)
⇒ `4x^2 - 4(1/4)x + 4(-1)`
⇒ 4x2 - x - 4
The quadratic polynomial is 4x2 - x - 4.
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 9x2 - 5 and verify the relation between the zeros and its coefficients.
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
4u2 + 8u
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
t2 – 15
If α and β are the zeros of the quadratic polynomial f(t) = t2 − 4t + 3, find the value of α4β3 + α3β4.
If the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.
If one zero of the quadratic polynomial f(x) = 4x2 – 8kx – 9 is negative of the other, find the value of k.
Find the zeroes of the polynomial f(x) = `2sqrt(3)x^2 - 5x + sqrt(3)` and verify the relation between its zeroes and coefficients.
Find the quadratic polynomial, sum of whose zeroes is `sqrt2` and their product is `(1/3)`.
What should be added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?
A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is
If \[\sqrt{5}\ \text{and} - \sqrt{5}\] are two zeroes of the polynomial x3 + 3x2 − 5x − 15, 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:
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`4x^2 + 5sqrt(2)x - 3`
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 factorization.
`21/8, 5/16`
If p(x) = x2 + 5x + 6, then p(– 2) is ______.
The zeroes of the polynomial p(x) = 2x2 – x – 3 are ______.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α2 + β2.
For the cubic equation \[ ax^{3} + bx^{2} + cx + d = 0 \] with roots \[ \alpha, \beta, \gamma \], the product of the roots \( \alpha\beta\gamma \) equals:
A quadratic equation whose roots are \[ \alpha, \beta \] can be written as:
