Let's tackle the given questions step-by-step.
### Question 5
#### Part (a)
The product of two consecutive positive numbers is 420. What are the numbers?
To find the two consecutive numbers whose product is 420, we can let the numbers be [tex]\( x \)[/tex] and [tex]\( x+1 \)[/tex]. So, the equation is:
[tex]\[ x(x + 1) = 420 \][/tex]
This can be rewritten as:
[tex]\[ x^2 + x - 420 = 0 \][/tex]
By solving this quadratic equation, we find that the two consecutive numbers are:
[tex]\[ x = 20 \][/tex]
[tex]\[ x + 1 = 21 \][/tex]
So, the two consecutive positive numbers are 20 and 21.
#### Part (b)
What should be subtracted from the numbers to get their product as 182?
Let the two numbers be [tex]\( 20 \)[/tex] and [tex]\( 21 \)[/tex]. We need to find a value [tex]\( x \)[/tex] such that subtracting it from these numbers gives us a product of 182. This means the numbers will be [tex]\( 20 - x \)[/tex] and [tex]\( 21 - x \)[/tex]. Hence, the equation is:
[tex]\[ (20 - x)(21 - x) = 182 \][/tex]
Expanding and simplifying, we get:
[tex]\[ 420 - 41x + x^2 = 182 \][/tex]
Rearranging this, we get a quadratic equation:
[tex]\[ x^2 - 41x + 238 = 0 \][/tex]
Solving this quadratic equation gives us:
[tex]\[ x = 34 \][/tex]
Therefore, the value that should be subtracted from both numbers to get their product as 182 is 34.
### Question 6
Solve the following:
Since there's no specified equation given in the statement "Solve the following", we likely need to solve a similar type of problem, but without a specific equation provided, I can't give a detailed solution here. Please provide the equation or problem you need to be solved for Question 6.
