The difference between two numbers is 9 and the product of the numbers is 162. Find the two numbers. Explain your working
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Let x be the larger number
x*(x-9) = 162
Rearrange and factor
x^2 - 9x -162 = 0
(x+9)(x - 18) = 0
Take it from there.
There will be two pairs of solutions.
To find the two numbers, let's assign variables to them. Let's call the larger number "x" and the smaller number "y".
According to the given information, the difference between the two numbers is 9. We can express this as an equation:
x - y = 9 ---(Equation 1)
Also, the product of the two numbers is 162. This can be represented as another equation:
x * y = 162 ---(Equation 2)
Now we have a system of two equations with two variables. We can solve them simultaneously using substitution or elimination method. Let's go with the substitution method here:
From Equation 1, solve for x in terms of y:
x = 9 + y
Substitute this value of x into Equation 2:
(9 + y) * y = 162
Simplify the equation:
9y + y^2 = 162
Rearrange the equation to make it equal to zero:
y^2 + 9y - 162 = 0
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 to simplify the equation:
(y + 18)(y - 9) = 0
Now we have two possibilities for y:
1) y + 18 = 0, which gives y = -18
2) y - 9 = 0, which gives y = 9
Now that we have the possible values for y, we can substitute them back into Equation 1 to find the corresponding values of x:
For y = -18:
x - (-18) = 9
x + 18 = 9
x = 9 - 18
x = -9
For y = 9:
x - 9 = 9
x = 9 + 9
x = 18
Therefore, the two numbers are (-9,9) or (18,9).
By symbols
A-B=9
AxB=162
so A=9+B
(9+B)B=162
9B+B^2=162
B^2+9B-162=0
so B=-18 or 9
correspondingly from AxB=162
A=-9 or 18
check against A-B=9
-9--18=9
18-9=9