Formation of a Quadratic Equation From It's Roots

Q. If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, then evaluate :

(i) α − β
(ii) $\frac{1}{\mathrm{\alpha }}-\frac{1}{\mathrm{\beta }}$
(iii) $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-2\mathrm{\alpha \beta }$
(iv) α²β − αβ²
(v) α⁴ + β⁴
(vi) $\frac{1}{a\mathrm{\alpha }+b}+\frac{1}{a\mathrm{\beta }+b}$
(vii) $\frac{\mathrm{\beta }}{a\mathrm{\alpha }+b}+\frac{\mathrm{\alpha }}{a\mathrm{\beta }+b}$
(viii) $a\left(\frac{{\mathrm{\alpha }}^2}{\mathrm{\beta }}+\frac{{\mathrm{\beta }}^2}{\mathrm{\alpha }}\right)+b\left(\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }}\right)$

Q.

Find the zeros of the quadratic polynomial √3x² - 8x + 4√3.

Q.

If $\alpha$, $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=2x^2-5x+7$. find a polynomial whose zeros are $2\alpha +3\beta$ and $3\alpha +2\beta$

Q.

Find a quadratic polynomial, the sum and product of whose zeros are $-3$ &$2$

Q.

Find the zeros of the polynomial $x^2-3x-m(m+3)$.

Q.

If one zero of polynomial 3x2-8x+2k+1 in seven times other find the zero and the value of the k

Q.

If α and β are the zeroes of the quadratic polynomial p(x) = x²– x – 2, find a polynomial whose zeroes
are 2α + 1 and 2β + 1.

Q.

If the sum of zeros of the quadratic polynomial p(x)=kx²+2x+3k is equal to their product find the value of k

Q.

If p and q are zeroes of polynomial f(x)=2x²-7x+3 find the value of p²+q²

Q.

What should be added to the quadratic polynomial x²-5x+4 so that 3 is a zero of the resulting polynomial ?

Q. If α and β are the zeros of the quadratic polynomial f(x) = x² − 1, find the quadratic polynomial whose zeros are $\frac{2\mathrm{\alpha }}{\mathrm{\beta }}\mathrm{and}\frac{2\mathrm{\beta }}{\mathrm{\alpha }}$.

Q. If α and β are the zeros of the quadratic polynomial p(x) = 4x² − 5x − 1, find the value of α²β + αβ².

Q.

If the sum of the squares of zeros of quadratic polynomial F (X)= x^2 - 8 x + K is 40, find k

Q.

If one zero of polynomial f(x) = (k² + 4)x² + 13x + 4k is reciprocal of the other, then k =

1. -2

2. -1

3. 2

4. 1

Q.

form a quadratic polynomial one of whose zeroes is 2+ root 5 and the sum of the zeroes is 4

Q. If α and β are the zeros of the quadratic polynomial p(s) = 3s² − 6s + 4, find the value of $\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{a}}+2\left(\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}\right)+3\mathrm{\alpha \beta }$.

Q. If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α − β = 8, find a quadratic polynomial have α and β as its zeros.

Q.

If α and $\frac{1}{\alpha }$ are the zeroes of

P(x)=4x²-2x+(k-4),

Find k.

Q.

if alpha and beta are zeroes of F(x)=X^2-P(x+1)-c show that (alpha+1)into (beta +1)=1-c

Q. If α and β are zeroes of the quadratic polynomial x² − 6x + a; find the value of 'a' if 3α + 2β = 20.

Q.

Find the quadratic polynomial whose zeroes are 3+√5 and 3-√5

Q.

If one zero of the quadratic polynomial x^{​​​​​​2} +:3x + k is 2, then find the value of k

Q.

If a and b are the zeroes of the polynomial f (X)=x² - 5x +k such that a-b =1 , find the value of k?

Q. If α and β are the zeros of the quadratic polynomial f(t) = t² − 4t + 3, find the value of α⁴β³ + α³β⁴.

Q. A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is

(a) x² − 9
(b) x² + 9
(c) x² + 3
(d) x² − 3

Q. If alpha and ß are zeroes of polynomial x²+6x+9, then form a quadratic polynomial whose zeroes are -alpha, -ß.

Q.

If sum of the zeroe of the polynomial Ky^2+2y-3K is twice their product of zeros then find the value of K

Q. For each of 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.
(i) $-\frac{8}{3},\frac{4}{3}$ (ii) $\frac{21}{8},\frac{5}{16}$ (iii) $-2\sqrt{3},-9$ (iv) $\frac{-3}{2\sqrt{5}},-\frac{1}{2}$Question

Q.

If 1 is the zero of the polynomial p(x) = ax² - 3(a - 1) x - 1, then find the value of a.

Q.

Find the quadratic polynomial whose sum & product of the zeroes r 21/8 & 5/16 respectively.Thanks