The discriminant D = b^2 - 4ac = 1^2 - 4(4)(1) = 1 - 16 = -15.
Since D = -15 < 0, the roots are Imaginary and unequal.
Given:
a - b = 7
ab = 6
The formula for a³ - b³ is:
a³ - b³ = (a - b)(a² + ab + b²)
= (a - b)((a - b)² + 3ab)
Substitute the values:
a³ - b³ = 7((7)² + 3(6))
= 7(49 + 18)
= 7 × 67
= 469
Calculate the cube root of 32768.
³√32768 = ³√32³ = 32
(-2)⁴ means (-2) × (-2) × (-2) × (-2) = 16
-2⁴ means -(2⁴) = -16
So:
(-2)⁴ > -2⁴
0 is called additive identity because a + 0 = a for any number 'a'.
It does not change the value of a number when added.
In a class interval (e.g., 10–20), the greatest value is called the upper class limit.
The lower class limit is the smallest value (e.g., 10), and the upper class limit is the largest value (e.g., 20).
The number of subsets of a set with n elements is 2^n.
Here, A = {a, b, c} has 3 elements, so:
Number of subsets = 2^3
= 8
The subsets are:
{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}
The given expression is:
a³ + 3ab(a + b) + b³
Factor the expression:
(a + b)(a² - ab + b²) + 3ab(a + b)
Combine like terms:
(a + b)(a² - ab + b² + 3ab)
Simplify:
(a + b)(a² + 2ab + b²)
Recognize the perfect cube:
(a + b)³
x + (1/x) = 8
Cube both sides:
(x + (1/x))³ = 8³
Expand the left side:
x³ + 3x²(1/x) + 3x(1/x)² + (1/x)³ = 512
x³ + 3x + 3(1/x) + (1/x)³ = 512
Rearrange:
x³ + (1/x)³ + 3(x + (1/x)) = 512
Substitute x + (1/x) = 8:
x³ + (1/x)³ + 3(8) = 512
x³ + (1/x)³ + 24 = 512
Subtract 24 from both sides:
x³ + (1/x)³ = 488
To find the square of the distance between two points A(x_1, y_1)A(x1,y1) and B(x2,y2):
AB^2=(x2−x1)^2+(y2−y1)^2 = (3 - 4)^2 + (5 - (-1))^2 = (-1)^2 + (6)^2 = 1 + 36 = 37=(3−4)^2+(5−(−1))^2
=(−1)^2+(6)^2
=1+36
=37