Relation between Roots and Coefficients of a Quadratic Equation

Q. $\text{If a and b are the zeroes of a polynomial}p{x}^2-5x+q,\text{then find the values of p}\text{and q, if a + b = ab = 10.}$

1. $5\text{and}\frac{1}{2}$
2. $\frac{1}{2}\text{and}5$
3. $5\text{and}2$
4. $10\text{and}1$

Q.

Find the sum and product of the roots of $2x^2$ – 3x + 5 = 0

Q.

Without solving, examine the nature of the roots of the equation $2x^2-6x+3=0$.

Q.

Find the quadratic polynomial whose sum of its zeroes (roots) is $-\frac{8}{5}$ and the product of the zeroes (roots) is $\frac{7}{5}$.

1. $14x^2+7x+5$

2. $5x^2+8x+7$

3. $2x^2-8x+7$

4. $5x^2-8x+7$

Q.

Find the sum and product of the roots of the quadratic equation: $x^2-5x+8=0$.

Q.

The ratio of the sum and the product of the roots of $5x^2–14x+8=0$ is___:___.

[Ratio in its reduced form]

Q.

If difference of reciprocal of roots of a quadratic equation $a{x}^2+bx+c=0$ is $\frac{1}{4}$, and difference of roots is 2. Then the value of c in the given equation is

1. 3

2. 4

3. 8

4. 6

Q. If $\alpha \text{and}\beta$ are the zeroes of $p\left(x\right)=8x^2+5x+k$ such that ${\alpha }^2+{\beta }^2+\alpha \beta =\frac{15}{4}$, then k =

Q.

The ratio of the sum and the product of the roots of $5x^2–14x+7=0$ is

___.

Q. If $x$ and $y$ are the roots of the equation $4k^2+5k+6=0$, then match the following with their values.

1. $-\frac{5}{4}$
2. $\frac{3}{2}$
3. $-\frac{5}{6}$

Q. Find the relation if the relation if the roots of $a{x}^2+bx+c=0$ are in the ratio $m:n$.

Q. If two roots of $x^3-9x^2+14x+24=0$ are in the ratio $3:2$ then the roots are

1. $6,4,-1$
2. $3,2,4$
3. $6,4,2$
4. $-6,4,1$

Q.

The sum of roots of the equation $x^2–8x+2=0$ is

___.

Q.

The sum of the roots of the equation $x^2–8x+2=0$ is

___.

Q.

The ratio of the sum and the product of the roots of $5x^2–14x+8=0$ is ___.

Q.

Find the sum of the roots of the equation $x^2–8x+2=0$.

1. -6

2. 2

3. -8

4. 8

Q. If p and q are the roots of $2x^2+3x+4$ = 0, them $p^{2}q+q^{2}p$ = .

1. -3
2. -4
3. 8
4. 16

Q. If k is the ratio of the roots of polynomial $x^2+px-q$, the value of $\frac{k}{1+k^2}$ is:

[1 mark]

1. $\frac{q}{p^2+2q}$
2. $\frac{q}{p^2+2q}$
3. $\frac{q}{p^2+2q}$
4. $\frac{-q}{p^2-2q}$

Q. State 'T' for true and 'F' for false.
I. If the square of the sum of the zeroes of a quadratic polynomial is 4 times the product of them, then the zeroes are equal in magnitude.
II. A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 4, then the linear term of the polynomial doesn't exist.
III. If the sum of the zeroes of a cubic polynomial is 5 and the sum of the product of two zeroes is 8, then the sum of the square of the zeroes will be 10.
IV. If one of the roots of the polynomial f(x) = $6x^2+13x+k$ is reciprocal of the other, then k is 6.

1. I - T II - F III - T IV - F
2. I - T II - T III - F IV - T
3. I - F II - T III - F IV - T
4. I - T II - F III - F IV - T

Q. Find the sum of the zeroes of the equation ${3x}^2+5x=2$.

Q. If the zeroes of the rational expression $(ax+b)(3x+2)$ are $-\frac{2}{3}$ and $\frac{1}{2}$, then $a+b=$

1. $4$
2. $0$
3. $-b$
4. None of these

Q. If $\alpha \&\beta$ are the zeros of $6x^2-x-2$ , then $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=?$

1. $\frac{-25}{12}$
2. $\frac{-12}{25}$
3. $\frac{24}{25}$
4. $\frac{25}{24}$

Q.

The ratio of the sum and the product of the roots of $5x^2–14x+8=0$ is

___.

Q.

Find a quadratic polynomial with the given numbers ​as the sum and product of its zeroes respectively.​

$\frac{-1}{4},\frac{1}{4}$

Q. Without solving, examine the nature of roots of the equation.

4$x^2$ – 4x + 1 = 0
[2 Marks]

Q. If k is the ratio of the roots of polynomial $x^2+px-q$, the value of $\frac{k}{1+k^2}$ is:

[1 mark]

1. $\frac{q}{p^2+2q}$
2. $\frac{q}{p^2+2q}$
3. $\frac{q}{p^2+2q}$
4. $\frac{-q}{p^2-2q}$

Q. Find the zeroes of $x^3+11x^2+23x-35$ If two of its zeros are $1$ and $-5$.

Q.

The ratio of the sum and the product of the roots of $5x^2–14x+8=0$ is

___.

Q. By using the formula, find the roots of the following quadratic equation, $(x+7)(x+3)=5(x+4)(x+2)+17x$

Q.

The sum of the roots of the equation $x^2–8x+2=0$ is

___.