"Sum of x and five" means (x + 5).
"Six times" means multiplying by 6.
So, the correct expression is 6(x + 5).
x: -3 < x < 6
This represents all numbers greater than -3 and less than 6.
A function f: A → B is into if it doesn't cover the entire codomain B, meaning there are elements in B that aren't mapped to by any element in A.
This happens when the range of f (the set of elements in B that are actually mapped to) is a proper subset of B.
Therefore, f is into if Range f ≠ B.
To factorize 4x² + 7x + 3:
Let's find two numbers that multiply to 4*3 = 12 and add to 7:
The numbers are 4 and 3 (since 4 + 3 = 7 and 4 * 3 = 12).
Now, rewrite the middle term:
4x² + 4x + 3x + 3
Factor by grouping:
4x(x + 1) + 3(x + 1)
(4x + 3)(x + 1)
The correct answer is (x + 1)(4x + 3).
The traveler covers:
First journey: x km
Second journey: y km
Third journey: x km
Total distance = x + y + x = 2x + y
To find the third proportion:
First term: a²b²
Second term: abc
The proportion is in the form:
a²b² : abc :: abc : x
To find x, set up a proportion equation:
a²b² / abc = abc / x
Simplify:
ab / c = abc / x
x = c²
Given cos θ = 4/5 and θ lies in the 4th quadrant.
In the 4th quadrant, cosine is positive, and sine is negative.
We know that sin² θ + cos² θ = 1
sin² θ = 1 - cos² θ
= 1 - (4/5)²
= 1 - 16/25
= 9/25
sin θ = ±√(9/25)
= ±3/5
Since θ lies in the 4th quadrant, sin θ is negative:
sin θ = -3/5
Given:
a² + b² + c² = 12
a + b + c = 5
We know that:
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
Expanding the equation:
5² = 12 + 2(ab + bc + ca)
25 = 12 + 2(ab + bc + ca)
Subtract 12 from both sides:
13 = 2(ab + bc + ca)
Divide by 2:
ab + bc + ca = 13/2
(44x+35y)+(34x−25y)
=(44x+34x)+(35y−25y)
=78x+10y
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by:
D = b^2 - 4ac
a = 1, b = 9, c = 2
D = 9^2 - 4(1)(2)
= 81 - 8
= 73