Answer:
The coefficient of the
x
2
-term is
80
.
Explanation:
First, we need to find the sixth row of Pascal's Triangle to determine the coefficient of the non-simplified terms of the expansion of the binomial expression. Remember that Pascal's Triangle begins with
1
as the first row and as the first and last entry of every other row. The middle terms of each row are obtained by adding the two terms from the row above.
1
/ \
1 1
/ \ / \
1 2 1
/ \ / \ / \
1 3 3 1
/ \ / \ / \ / \
1 4 6 4 1
/ \ / \ / \ / \ / \
1 5 10 10 5 1
Remember that in this case,
(
x
+
2
)
5
,
a
=
x
and
b
=
2
. The binomial theorem tells us that in the expansion, the terms will follow this pattern:
a
5
b
0
+
a
4
b
1
+
a
3
b
2
+
a
2
b
3
+
a
1
b
4
+
a
0
b
5
Combining that with the coefficients from Pascal's Truangle gives us this expansion of
(
x
+
2
)
5
:
#1(x^5)(2^0) + 5(x^4)(2^1) + 10(x^3)(2^2) + 10(x^2)(2^3) + 5(x^1)(2^4) + 1(x^0)(2^5)
The fourth term is the
x
2
-term, so we need to simplify it:
#10(x^2)(2^3) = 10(x^2)(8) = 80x^2
So, the coefficient of the
x
2
-term is
80
.
Answer: Either

or

depending on your teacher's preference. See the note at the bottom for more info.
=====================================
Let x be the speed limit. It is simply a placeholder. Think of it as a box with the number inside. We don't know what the number is, but we know that the largest it can be is 55. It cannot be larger than 55.
So x can be equal to 55 or it can be smaller
Put another way, x is less than or equal to 55 which is written as

(on the keyboard you would type " x <= 55 " without quotes).
---------------------------------------
Note: since negative speeds make no sense, we also must imply that x can't be negative. Your teacher may want you to write this in to clearly state it. If so, then you would also have

which when combined with the first inequality, you get this compound inequality

(basically saying "x is some speed between 0 and 55 mph"). This very clearly states the boundaries on what x can be. Though your teacher may want you to stick to the first format as its simpler.
Answer:
A bus travels with a constant speed of 48 miles per hour. How long will it take to travel 60 miles?
Step-by-step explanation:
Mostly it's 1.25 hours </3 Since distance = rate times , or d = rt, then we have
t = d/r = 60 miles / 48 mph = 1.25 hours = 1 hour 15 minutes
either 1 hour 15 mins or 1hour and 25 mins.
The unit rate for the nearest cent per once for each size is 0.24 : 0.19