Advertisements
Advertisements
प्रश्न
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.
`(-3)/(2sqrt(5)), -1/2`
Advertisements
उत्तर
Sum of the zeroes = `-3/2 sqrt(5)x`
Product of the zeroes = `- 1/2`
P(x) = x2 – (Sum of the zeroes) + (Product of the zeroes)
Then, P(x) = `x^2 - (-3/2 sqrt(5)x) - 1/2`
P(x) = `2sqrt(5)x^2 + 3x - sqrt(5)`
Using splitting the middle term method,
`2sqrt(5)x^2 + 3x - sqrt(5)` = 0
`2sqrt(5)x^2 + (5x - 2x) - sqrt(5)` = 0
`2sqrt(5)x^2 - 5x + 2x - sqrt(5)` = 0
`sqrt(5)x (2x + sqrt(5)) - (2x + sqrt(5))` = 0
`(2x + sqrt(5))(sqrt(5)x - 1)` = 0
`\implies` x = `1/sqrt(5), -sqrt(5)/2`
APPEARS IN
संबंधित प्रश्न
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
Verify that the numbers given along side of the cubic polynomials below are their zeroes. Also verify the relationship between the zeros and the coefficients.
g(x) = x3 – 4x2 + 5x – 2; 2, 1, 1
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α4 + β4
If α and β are the zeros of the quadratic polynomial f(x) = 6x2 + x – 2, find the value of `alpha/beta + beta/alpha`.
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 squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.
Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.
Find the zeroes of the quadratic polynomial` (x^2 ˗ 5)` and verify the relation between the zeroes and the coefficients.
Find the zeroes of the quadratic polynomial (3x2 – x – 4) and verify the relation between the zeroes and the coefficients.
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then α2 + β2 + γ2 =
What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?
If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then
Case Study -1
The figure given alongside shows the path of a diver, when she takes a jump from the diving board. Clearly it is a parabola.
Annie was standing on a diving board, 48 feet above the water level. She took a dive into the pool. Her height (in feet) above the water level at any time ‘t’ in seconds is given by the polynomial h(t) such that h(t) = -16t2 + 8t + k.
The zeroes of the polynomial r(t) = -12t2 + (k - 3)t + 48 are negative of each other. Then k 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:
4x2 – 3x – 1
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`
Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.
