Advertisements
Advertisements
प्रश्न
Find the zeroes of the quadratic polynomial 4x2 – 4x + 1 and verify the relation between the zeroes and the coefficients.
Find the zeroes of the polynomial 4x2 – 4x + 1 and verify there rationship between the zeroes and the coefficients.
Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:
4x2 – 4x + 1
Advertisements
उत्तर १
4x2 – 4x + 1 = 0
⇒ (2x2) – 2(2x)(1) + (1)2 = 0
⇒ (2x – 1)2 = 0 ...[∵ a2 – 2ab + b2 = (a – b)2]
⇒ (2x – 1)2 = 0
⇒ `x = 1/2` or `x = 1/2`
Sum of zeroes = `1/2 + 1/2`
= 1
= `1/1`
= `(-("Coefficient of" x))/(("Coefficient of" x^2))`
Product of zeroes = `1/2 xx 1/2`
= `1/4`
= `("Constant term")/(("Coefficient of "x^2))`
उत्तर २
Given, Polynomial is 4x2 – 4x + 1 ...(i)
⇒ 4x2 – 2x – 2x + 1
⇒ 2x(2x – 1) – 1(2x – 1)
⇒ (2x – 1) (2x – 1)
⇒ `x = 1/2`
Hence, zeroes of a given polynomial is `x = 1/2`
On comparing equation (i) with ax2 + bx + c = 0,
We get a = 4, b = – 4 and c = 1
Now, the sum of zeroes = `(-b)/a = (-(-4))/4 = 1`
Product of zeroes = `c/a = 1/4` which matches with:
Sum of the zero = `1/2 + 1/2`
= `2/2`
= 1
Product of the zero = `1/2 xx 1/2`
= `1/4`
APPEARS IN
संबंधित प्रश्न
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
3x2 – x – 4
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`q(x)=sqrt3x^2+10x+7sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate :
`a(α^2/β+β^2/α)+b(α/β+β/α)`
If α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 6s + 4, find the value of `alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta`
If α and β are the zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.
If (x+a) is a factor of the polynomial `2x^2 + 2ax + 5x + 10`, find the value of a.
Verify that 3, –2, 1 are the zeros of the cubic polynomial p(x) = (x3 – 2x2 – 5x + 6) and verify the relation between it zeros and coefficients.
Find a cubic polynomial whose zeroes are `1/2`, 1 and –3.
By actual division, show that x2 – 3 is a factor of 2x4 + 3x3 – 2x2 – 9x – 12.
If 3 and –3 are two zeroes of the polynomial (x4 + x3 – 11x2 – 9x + 18), find all the zeroes of the given polynomial.
If α, β are the zeroes of the polynomial f(x) = x2 + x – 2, then `(α/β - α/β)`.
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 ______.
A quadratic polynomial, whose zeroes are –3 and 4, 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 ______.
If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
3x2 + 4x – 4
The only value of k for which the quadratic polynomial kx2 + x + k has equal zeros is `1/2`
Find the sum and product of the roots of the quadratic equation 2x2 – 9x + 4 = 0.
