Nature of roots and formation of quadratic equations - One Line Questions
1.
For the equation x^2 + 5x + 6 = 0, the roots are: —
-2 and -3
2.
If alpha and beta are the roots of x^2 - 7x + 10 = 0, find alpha + beta. —
7
3.
If the roots of the equation x^2 + bx + c = 0 are alpha and beta, then the sum of roots alpha+beta is related to coefficients as: —
-b
4.
If the roots of the equation x^2 + bx + c = 0 are alpha and beta, then the product of roots alpha*beta is related to coefficients as: —
c
5.
If one root of the equation x^2 - 3x + k = 0 is 2, find the other root. —
1
6.
If the roots of x^2 + kx + 1 = 0 are real and equal, find the value of k. —
2
7.
If alpha and beta are the roots of x^2 - 2x + 1 = 0, find alpha*beta. —
1
8.
If one root of the equation kx^2 - 3x + 2 = 0 is 1, find the value of k. —
1
9.
If one root of x^2 - 3x + k = 0 is 1, find the other root. —
2
10.
If alpha and beta are the roots of x^2 - 5x + 6 = 0, find the value of (alpha+1)(beta+1). —
10
11.
If one root of the equation x^2 - 7x + k = 0 is 4, find the value of k. —
12
12.
If alpha and beta are the roots of x^2 - 5x + 6 = 0, find alpha^2 + beta^2. —
13
13.
Given the quadratic equation x^2 + 4x + k = 0 has equal roots, find the value of k. —
4
14.
What is the sum of the roots of the quadratic equation 3x^2 - 6x + 5 = 0? —
2
15.
Consider the equation x^2 + 2x + 3 = 0. The sum of squares of its roots (alpha^2 + beta^2) is: —
-2
16.
If the roots of x^2 - kx + 4 = 0 are real and equal, find the value of k. —
4
17.
If one root of the quadratic equation x^2 - kx + 4 = 0 is 2, what is the value of k? —
5
18.
If one root of the equation x^2 - 5x + c = 0 is 3, find the value of c. —
6
19.
What are the roots of the quadratic equation x^2 - 6x + 9 = 0? —
3 and 3
20.
What is the product of the roots of the quadratic equation 2x^2 + 5x - 7 = 0? —
-7/2
21.
If alpha and beta are roots of x^2 - 5x + 6 = 0, find the value of 1/alpha + 1/beta. —
5/6
22.
Form a quadratic equation with roots 1/2 and 1/3. —
6x^2 - 5x + 1 = 0
23.
If the roots of x^2 + 6x + k = 0 are real and equal, find the value of k. —
9
24.
If the roots of ax^2 + bx + c = 0 are reciprocal of each other, then: —
a = c
25.
If the roots of ax^2 + bx + c = 0 are opposite to each other, then: —
b = 0
26.
If the roots of x^2 + ax + b = 0 are real and distinct, then: —
a^2 - 4b > 0
27.
If alpha and beta are the roots of ax^2 + bx + c = 0, then the equation whose roots are alpha+k and beta+k is: —
a(x-k)^2 + b(x-k) + c = 0
28.
The quadratic equation x^2 - (alpha + beta)x + alpha*beta = 0 has roots: —
alpha and beta
29.
What is the condition for the quadratic equation ax^2 + bx + c = 0 to have real roots? —
b^2 - 4ac >= 0
30.
If the roots of the quadratic equation ax^2 + bx + c = 0 are alpha and beta, then the sum of the roots is given by: —
-b/a
31.
For a quadratic equation ax^2 + bx + c = 0 with roots alpha and beta, the product of the roots is: —
c/a
32.
If alpha and beta are the roots of ax^2 + bx + c = 0, then the equation whose roots are 1/alpha and 1/beta is: —
cx^2 + bx + a = 0
33.
If the roots of the quadratic equation ax^2 + bx + c = 0 are real and distinct, what is the condition on the discriminant (D = b^2 - 4ac)? —
D > 0
34.
For a quadratic equation ax^2 + bx + c = 0, if the roots are real and equal, the discriminant is: —
D = 0
35.
If the roots of the quadratic equation ax^2 + bx + c = 0 are imaginary, then the discriminant (D) satisfies: —
D < 0
36.
If the roots of ax^2 + bx + c = 0 are real and distinct, then b^2 - 4ac is: —
Greater than 0
37.
If the roots of x^2 - 6x + k = 0 are real and distinct, what is the range of k? —
k < 9
38.
If the roots of x^2 + 4x + k = 0 are imaginary, find the range of k. —
k > 4
39.
If one root of x^2 + px + q = 0 is 2, and the other root is 3, what are the values of p and q? —
p = -5, q = 6
40.
If the roots of the equation x^2 - px + q = 0 are alpha and beta, then alpha^2 + beta^2 = ? —
p^2 - 2q
41.
Consider the quadratic equation x^2 - 5x + 6 = 0. What is the nature of its roots? —
Real and distinct
42.
Determine the nature of the roots for the equation 2x^2 + 3x + 4 = 0. —
Imaginary
43.
For the equation 4x^2 - 12x + 9 = 0, the nature of the roots is: —
Real and equal
44.
What is the nature of the roots of the equation x^2 + x + 1 = 0? —
Imaginary
45.
If the discriminant of ax^2 + bx + c = 0 is positive, the roots are: —
Real and distinct
46.
If the roots of x^2 + px + 12 = 0 are equal, find the value of p. —
sqrt(48)
47.
If the roots of x^2 + bx + c = 0 are alpha and beta, then the equation whose roots are alpha^2 and beta^2 is: —
x^2 - (b^2-2ac)x + c^2 = 0
48.
Form a quadratic equation whose roots are -2 and 5. —
x^2 - 3x - 10 = 0
49.
Form a quadratic equation whose roots are 2+sqrt(3) and 2-sqrt(3). —
x^2 - 4x + 1 = 0
50.
Form a quadratic equation whose roots are 2 and 3. —
x^2 - 5x + 6 = 0