Quadratic equations in real and complex systems - One Line Questions
1.
If one root of the equation (k+1)x^2 - 3(k+4)x + 4(k+3) = 0 is 1, then find the value of k. —
-3
2.
For the quadratic equation ax^2 + bx + c = 0, what is the product of the roots (αβ)? —
c/a
3.
If α and β are the roots of ax^2 + bx + c = 0, then the value of (α + 1/β)(β + 1/α) is: —
(b^2 - ac)/a^2
4.
The number of real solutions of the equation x^2 + 2|x| + 1 = 0 is: —
0
5.
The number of real roots of the equation x^2 + 1/x^2 - (x + 1/x) - 2 = 0 is: —
2
6.
The number of real solutions for the equation |x^2 - 1| = |x - 1| is: —
3
7.
If α and β are the roots of x^2 + x + 1 = 0, then the value of α^3 + β^3 is: —
-2
8.
If α and β are the roots of x^2 - 3x + 1 = 0, then the value of (α^2 + 1)(β^2 + 1) is: —
3
9.
If α and β are the roots of x^2 - (k-2)x + (k+1) = 0, find k such that α+β = αβ. —
2
10.
The roots of the equation (x^2 - 5x + 5)^(x^2 - 11x + 30) = 1 are: —
1, 2, 3, 5, 6
11.
If α and β are roots of x^2 + 2x + 4 = 0, then α^3 + β^3 is: —
-32
12.
If the roots of the equation x^2 - 6x + k = 0 are α and β, and α^2 + β^2 = 20, then k is: —
2
13.
What are the roots of the quadratic equation x^2 + 4 = 0? —
2i, -2i
14.
If one root of the quadratic equation x^2 - 5x + k = 0 is 2, what is the value of k? —
6
15.
The minimum value of the expression x^2 - 4x + 7 is: —
3
16.
What are the roots of the quadratic equation x^2 - 6x + 13 = 0? —
3 + 2i, 3 - 2i
17.
The values of x satisfying the equation x^2 - |x| - 6 = 0 are: —
3, -2
18.
If one root of the equation ax^2 + bx + c = 0 is 3 times the other, then: —
3b^2 = 16ac
19.
What is the value of k for which the equation x^2 - kx + 4 = 0 has equal roots? —
±4
20.
If α and β are the roots of x^2 - 2x + 4 = 0, then the value of α^3 is: —
8
21.
If one root of the equation ax^2 + bx + c = 0 is 0, then: —
c = 0
22.
If the roots of the equation ax^2 + bx + c = 0 are reciprocal to each other, then: —
a = c
23.
If the roots of the equation ax^2 + bx + c = 0 are opposite to each other, then: —
b = 0
24.
If the sum of the roots of the equation ax^2 + bx + c = 0 is equal to the product of the roots, then: —
a = c
25.
If the roots of the equation x^2 + ax + b = 0 are α and β, and the roots of x^2 + cx + d = 0 are αβ and 1/(αβ), then: —
b = 1/d
26.
If α and β are the roots of ax^2 + bx + c = 0, then the quadratic equation whose roots are α^2 and β^2 is: —
a^2x^2 - (b^2 - 2ac)x + c^2 = 0
27.
If the roots of the equation x^2 + ax + b = 0 are α and β, and the roots of x^2 + cx + d = 0 are α + k and β + k, then: —
c-a = k
28.
If α and β are the roots of ax^2 + bx + c = 0, then the quadratic equation whose roots are α + 1 and β + 1 is: —
a(x-1)^2 + b(x-1) + c = 0
29.
If α, β are roots of ax^2 + bx + c = 0, then the equation whose roots are α+k and β+k is given by: —
a(x-k)^2 + b(x-k) + c = 0
30.
What is the standard form of a quadratic equation? —
ax^2 + bx + c = 0
31.
If the equation x^2 - bx + 1 = 0 has real roots, then: —
b < -2 or b > 2
32.
The equation ax^2 + bx + c = 0 has real roots if: —
b^2 - 4ac ≥ 0
33.
For the quadratic equation ax^2 + bx + c = 0, what is the sum of the roots (α + β)? —
-b/a
34.
If α and β are the roots of ax^2 + bx + c = 0, then the quadratic equation whose roots are 1/α and 1/β is: —
cx^2 + bx + a = 0
35.
If the roots of the equation ax^2 + bx + c = 0 are in the ratio m:n, then: —
mn b^2 = (m+n)^2 ac
36.
If one root of the quadratic equation x^2 - px + q = 0 is sin(30°) and the other is cos(60°), find p and q. —
p=1, q=1/4
37.
If the roots of the equation (p^2 + q^2)x^2 - 2(p+q)x + 2 = 0 are real, then: —
p=-q
38.
If the roots of the quadratic equation px^2 + qx + r = 0 are equal, what is the condition on its coefficients? —
q^2 = 4pr
39.
What is the nature of the roots of the equation x^2 + ix + 1 = 0? —
Complex and not conjugates
40.
The roots of the quadratic equation (a+b)x^2 + 2(a+b)x + c = 0 are: —
Real and equal
41.
If the discriminant (Δ = b^2 - 4ac) of a quadratic equation with real coefficients is zero (Δ = 0), what can be said about its roots? —
The roots are real and equal
42.
If the discriminant (Δ = b^2 - 4ac) of a quadratic equation with real coefficients is negative (Δ < 0), what can be said about its roots? —
The roots are complex conjugates
43.
If the discriminant (Δ = b^2 - 4ac) of a quadratic equation with real coefficients is positive (Δ > 0), what can be said about its roots? —
The roots are real and distinct
44.
Consider the quadratic equation ax^2 + bx + c = 0. If the coefficients a, b, and c are complex numbers, what can be said about the nature of the roots? —
The roots are not necessarily complex conjugates.
45.
For a quadratic equation ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0, what does the discriminant (Δ) represent? —
The nature of the roots
46.
The equation x^2 + |x| + 1 = 0 has: —
No real roots
47.
The equation x^2 - |x| - 2 = 0 has: —
Four real roots
48.
Which formula gives the roots of a quadratic equation ax^2 + bx + c = 0? —
x = [-b ± √(b^2 - 4ac)] / 2a
49.
The quadratic equation that has roots 2+i and 2-i is: —
x^2 - 4x + 5 = 0
50.
Find the quadratic equation whose roots are 3 and 4. —
x^2 - 7x + 12 = 0