The difference between two numbers is 6. Their product is 2,016. What are the two numbers.
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L-S=6
LS=2016
(6+S)(S)=2015
multiply out, solve the quadratic.
another way. sqrt 2016=44.89
so try numbers on each side of 45+-3
42, 48 difference six, product...
a = First number
b = Second number
The difference between two numbers is 6.
This mean :
a - b = 6
Their product is 2016.
This mean :
a * b = 2016
Now :
a - b = 6 Add b to both sides
a - b + b = 6 + b
a = 6 + b
a = b + 6
Replace this value in expression :
a * b = 2016
( b + 6 ) * b = 2016
b ^ 2 + 6 b = 2016 Subtract 2016 to both sides
b ^ 2 + 6 b - 2016 = 2016 - 2016
b ^ 2 + 6 b - 2016 = 0
The solutions are :
b = - 48 and b = 42
For b = - 48
a = b + 6 = - 48 + 6 = - 42
For b = 42
a = b + 6 = 42 + 6 = 48
The solutions are :
- 48 , - 42
and
42 , 48
the real answer is 11 and 5
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To find the two numbers, let's break it down step by step:
Step 1: Let's assign variables to the two numbers. Let's call them "x" and "y".
Step 2: We know that the difference between the two numbers is 6, which can be written as an equation: x - y = 6.
Step 3: We also know that their product is 2016, which can be written as another equation: x * y = 2016.
Step 4: Now we have a system of two equations with two variables:
x - y = 6 ... Equation 1
x * y = 2016 ... Equation 2
Step 5: To solve the system of equations, we can use substitution or elimination method. In this case, let's use the substitution method.
Step 6: Solve Equation 1 for x. Add y to both sides of the equation: x = y + 6.
Step 7: Substitute the value of x in Equation 2: (y + 6) * y = 2016.
Step 8: Simplify the equation: y^2 + 6y = 2016.
Step 9: Rearrange the equation: y^2 + 6y - 2016 = 0.
Step 10: Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring.
Step 11: The factored form of the equation is: (y + 56)(y - 36) = 0.
Step 12: Set each factor to zero and solve for y:
y + 56 = 0 -> y = -56
y - 36 = 0 -> y = 36
Step 13: We have two possible values for y: -56 and 36.
Step 14: Now substitute the values back into Equation 1 to solve for x:
x - (-56) = 6 -> x + 56 = 6 -> x = -50
x - 36 = 6 -> x = 42
Step 15: We have two possible values for x: -50 and 42.
Therefore, the two numbers that satisfy both conditions are -50 and 36, or 42 and -36.