Relationship Between Roots and Coefficients of Quadratic Equation

Q.

if the sum of the zeroes of $9x^2-kx+2$ is $\frac{11}{9}$ find the value of k

1. -1

2. 1

3. -11

4. 11

Q.

Find a quadratic polynomial whose zeros are $-3$ and $2$.

Q.

The value of k such that $3x^2+2kx+x-k-5$ have the sum of the zeroes as half of their product,

1. 4

2. 2

3. 3

4. 1

Q. Given that one of the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$ is zero, the product of the other two zeroes is:

1. \N
2. $\frac{c}{a}$
3. $\frac{-c}{a}$
4. $\frac{-b}{a}$

Q.

Find a quadratic polynomial whose zeroes are $-4$ and $2$.

Q.

If one of the zeroes of the polynomial $g\left(x\right)=(k^2+4)x^2+13x+4k$ is reciprocal of the other, then value of k is,

1. 0

2. 1

3. 2

4. 3

Q.

If one of the zeroes of the polynomial g(x) = $(k^2+4)x^2+13x+4k$ is reciprocal of the other then k is

1. 2

2. 3

3. 4

4. 1

Q. If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2-2x+3$, find a polynomial whose roots are
(i) $\alpha +2,\beta +2$
(ii) $\frac{\alpha -1}{\alpha +1},\frac{\beta -1}{\beta +1}.$

Q. If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial such that $\alpha +\beta =24$ and $\alpha -\beta =8$ find a quadratic polynomial having a $\alpha$ and $\beta$ as its zeros.

Q. If one of the zero of polynomial $3x^2-8x+2k+1$ is seven times the other, find the value of k.

Q. If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $x^2+5x-k$ and $\alpha -\beta =1$, then find the value of $k$.

Q. Find the zeros of the quadratic polynomial $x^2+5x+6$ and verify the relationship between the zeros and the coefficients.

Q. Given that one of the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$ is zero, the product of the other two zeroes is:

1. \N
2. $\frac{c}{a}$
3. $\frac{-c}{a}$
4. $\frac{-b}{a}$

Q. Find a quadratic polynomial whose zeroes are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$. If $\alpha$ and $\beta$ are the zeroes of $2x^2-3x+5$

Q. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=x^2-(\sqrt{5}-1)x-(\sqrt{5}+1)$, then the value of $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ is ____________.

1. $3+\sqrt{5}$
2. $3-\sqrt{5}$
3. $\sqrt{5}-3$
4. $-3-\sqrt{5}$

Q. If the zeros of polynomial $k^2{x}^2-16(k+x)+k+2,(k<0)$ are reciprocal of each other, then the value of $k$ is:

[1 mark]

1. -1
2. 1
3. 0
4. 2

Q.

zeroes of $x^2+kx+k$

1. cant be both negative

2. always unequal

3. cant be both positive

4. always equal

Q.

What are the possible rational roots of $x^3-5x^2-4x+20=0$ and then determine the rational roots?

Q. Given that one of the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$ is zero, the product of the other two zeroes is:

1. \N
2. $\frac{c}{a}$
3. $\frac{-c}{a}$
4. $\frac{-b}{a}$

Q.

If the sum of the zeroes of $k{x}^2-5x+2$ is $\frac{5}{2}$ find the value of k

1. 

2. 1

3. 2

4. 5

Q. If $\alpha \text{and}\beta$ are the zeroes of $p\left(x\right)=4x^2+10x+k$ such that ${\alpha }^2+{\beta }^2+\alpha \beta =\frac{21}{4}$, then k =

1. 12
2. 4
3. 2
4. - 12

Q. Find the sum of roots and product of roots respectively for the polynomial :
$2x^2+x-5=0$

1. $-\frac{1}{2},\frac{5}{2}$
2. 2, 5
3. $-\frac{1}{2},-\frac{5}{2}$
4. $-\frac{1}{2},\frac{5}{2}$

Q. If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $x^2-3x+2$, then a quadratic polynomial whose zeros are $\frac{1}{2\alpha +\beta }$ and $\frac{1}{2\beta +\alpha }$ is?

1. $20x^2+9x+1$
2. $20x^2-9x-1$
3. $20x^2-9x+1$
4. $20x^2+9x-1$

Q. Find the polynomial whose zeros are $\alpha =1/3$ and $\beta =1/2$

Q. The sum and product of zeroes of a quadratic polynomial $a{x}^2+bx+c$ respectively are and .

1. -ba
2. ca
3. ba
4. -ca

Q. Find the value of p, for which one root of the quadratic $p{x}^2-14x+8=0$ is 6 times the other.

1. 9
2. 12
3. 6
4. 3