Sum and Product of Roots of a Quadratic Equation

Q.

Find the zeros of the following quadratic polynomials $4s^2-4s+1$ and verify the relationship between the zeros and the coefficients.

Q.

Find the quadratic polynomial, sum of whose zeros is $\left(\frac{5}{2}\right)$ and their product is $1$. Hence, find the zeros of the polynomial?

Q.

If $x=\frac{2}{3}andx=-3$ are the roots of the quadratic equation $a{x}^2+7x+b=0$ then find the values of a and b.

Q.

$\text{Find the zeroes of the polynomial}p\left(x\right)=4x^2-4x+1\text{and verify the relationship between the zeroes and the coefficients.}$

Q.

If the sum of the zeros of the quadratic polynomial $k{x}^2+2x+3k$ is equal to the product of its zeros then k = ?

(a) $\frac{1}{3}$ (b) $\frac{-1}{3}$ (c) $\frac{2}{3}$ (d) $\frac{-2}{3}$

Q.

The zeros of the quadratic polynomial $x^2+88x+125$ are

(a) both positive (b) both negative

(c) one positive and one negative (d) both equal

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) $\frac{-3}{2\sqrt{5}},-\frac{1}{2}$

Q.

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coeffients :

$5x^2-4-8x$

Q.

Find the quadratic polynomial, the sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial.

Q.

If the product of the roots of $x^2–3x+k=10$ is –2, then find the value of k.

1. -2

2. 8

3. 12

4. 6

Q. Question 2 (i)
Are the following statements ‘True’ or False’? Justify your answer.
i) If the zeroes of a quadratic polynomial $a{x}^2+bx+c$ are positive then a, b and c all have the same sign.

Q. Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
$t^2-15$

Q.

The sum and the product of the zeros of a quadratic polynomial are 3 and -10 respectively. The quadratic polynomial is

(a) $x^2-3x+10$ (b) $x^2+3x-10$ (c) $x^2-3x-10$ (d) $x^2+3x+10$

Q.

Find the zeroes of the following polynomial by factorization method: $3x^2+4x-4$.

Q. Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
$4s^2-4s+1$

Q.

If one zero of the polynomial $\left(a^2+2\right)x^2+10x+3a$ is the reciprocal of the other. find $a$.

Q.

If the sum of the roots of a quadratic equation is 5 and the product of the roots is also 5, then the equation can be:

1. 

2. 

3. 

4. 

Q. The roots of the quadratic equation $3x^2+2x-1=0$ are .

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

Q.

For the same amount of work, A takes 6 hours less than B. If together they complete the work in 13 hours 20 minutes, find how much time will B alone take to complete the work?

1. 40 hrs

2. 10 hrs

3. 15 hrs

4. 30 hrs

Q. Question 1 (iv)
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.
iv) $=\frac{-3}{2\sqrt{5}},-\frac{1}{2}$

Q.

If the product of the roots of $x^2–3x+k=10$ is –2, then find the value of k.

1. -2

2. 8

3. 12

4. 6

Q.

Question 1 (v)
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

$t^2-15$

Q.

The sum of two numbers is 48 and their product is 432 Find the numbers.

Q.

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coeffients :

$2x^2-11x+15$

Q.

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients :

$2\sqrt{3}{x}^2-5x+\sqrt{3}$

Q. Question 1 (x)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials

(x) $7y^2–\frac{11}{3}y-\frac{2}{3}$

Q.

Question 1 (ii)
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

$4s^2-4s+1$

Q.

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coeffients :

$4x^2-4x-3$

Q.

Find the values of k for which the given quadratic equation has real and distinct roots:
$\left(i\right)k{x}^2+2x+1=0\left(ii\right)k{x}^2+6x+1=0$

Q. Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
$6x^2-3-7x$