2b or not 2b Trivia Quiz

Answer: All values of b

This is quite straightforward. 3b - b = 2b , so the expression is equal to 2b for all values of b.

Answer: All values of b

This appears to be a bit more convoluted.

The innermost expression in the brackets is sq rt 16b, which is asking for the square root of 16b,
or a number which, when squared is equal to 16b.

The next bracket requires that number to be squared, so we get 16b.

Then the division needs to be done. 16b / 8 = 2b. This will be true for all values of b.

Answer: All values of b

This can be broken down into three parts.

b x 2^5 = b x 32, which is usually written as 32b.

4^2 = 16.

32b / 16 = 2b, so the original expression is true for all values of b.

Answer: All values of b

This also needs solving in separate stages.

((3 x 4) x b) = (12 x b) = 12b

((2 x 5) x b) = (10 x b) = 10b

Then the subtraction is straightforward - 12b - 10b = 2b, which is true for all values of b.

Answer: Some values of b

If b^2 is to be equal to 2b, there needs to be a solution to the equation b^2 = 2b.

b^2 = 2b is the same as b^2 - 2b = 0.
This can be factorised as b x (b - 2) = 0.
Since any number multiplied by 0 is equal to 0, this will be true if either of the factors are 0.
That is either b = 0 or (b - 2) = 0. In the second case, that would mean b = 2.

So the original expression is equal to 2b only if b = 0 or if b = 2.

Answer: Some values of b

If b x (b - 1) is to be equal to 2b, there needs to be a solution to the equation b x (b - 1) = 2b.

b x (b - 1) = 2b is the same as b^2 - b = 2b, or b^2 - 3b = 0.
This can be factorised as b x (b - 3) = 0.
Since any number multiplied by 0 is equal to 0, this will be true if either of the factors are 0.
That is either b = 0 or (b - 3) = 0. In the second case, that would mean b = 3.

So the original expression is equal to 2b only if b = 0 or if b = 3.

Answer: Some values of b

If (b + 1)^2 / b is to be equal to 2b, there needs to be a solution to the equation (b + 1)^2 / b = 2b.

(b + 1)^2 / b = 2b is the same as (b^2 + 2b + 1) / b = 2b.
Multiplying both sides of the equation by b gives b^2 + 2b + 1 = 2b^2.
That is the same as b^2 - 2b - 1 = 0.
Using the standard formula for the solution to a quadratic equation gives the solution as
b = (2 +- sq rt (4 + 4)) / 2
or b = (2 +- sq rt 8) / 2
or b = (2 +- 2 x sq rt (2)) / 2
or b = 1 +- sq rt 2.

That leads to two solutions - either b is approximately equal to 2.41 or b is approximately equal to
-0.41.

So the original expression is equal to 2b for two values of b.

Answer: Some values of b

If (b + 1) / b is to be equal to 2b, there needs to be a solution to the equation (b + 1) / b = 2b.

(b + 1) / b = 2b is the same as b + 1 = 2b^2.
That is the same as 2b^2 - b - 1 = 0.
This can be factorised to get (2b + 1) x ( b - 1) = 0.
So either 2b + 1 must be 0 or b - 1 must be zero to get a solution to the equation.
If 2b + 1 = 0, then b = -1 / 2.
If b - 1 = 0, then b = 1.

So the original expression is equal to 2b for two values of b.

Answer: Only if b = 0

If 4b - b is to be equal to 2b, there needs to be a solution to the equation 4b - b = 2b.
That equation is the same as 4b - b - 2b = 0.
This can be simplified as b = 0.

Answer: Only if b = 0

The expression b + b/2 + b/3 + b/4 can be re-written as (12b / 12) + (6b / 12) + (4b / 12) + (3b / 12)
or (12b + 6b + 4b + 3b) / 12 or 25b / 12.

If the original expression is to be equal to 2b, there must be a solution to the equation 25b / 12 = 2b,
which is the same as 25b = 24b, or 25b - 24b = 0, or b = 0.

Answer: Only if b = 0

The innermost expression in the brackets is sq rt 3b, which is asking for the square root of 3b,
or a number which, when squared is equal to 3b.
The next bracket requires that number to be squared, so we get 3b.

So if the original expression is to be equal to 2b, there must be a solution to the equation 3b / 3 = 2b.
That is the same as 3b = 6b, which is the same as 6b - 3b = 0 or 3b = 0,
which leads to the solution b = 0.

Answer: Only if b = 0

((3 x 5) x b) - ((2 x 4) x b) can be simplified as 15b - 8b, which is equal to 7b.

The only solution to 7b equalling 2b is when b = 0.