Find two numbers (exactly) whose product is 10 and whose sum is 18.
a = first number
b = second number
Conditions :
a * b = 10
a + b = 18
a + b = 18 Subtract a to both sides
a + b - a = 18 - a
b = 18 - a
a * b = 10
a * ( 18 - a ) = 10
18 a - a ^ 2 = 10
- a ^ 2 + 18 a = 10 Multiply both sides by - 1
a ^ 2 - 18 a = - 10 [ Add ( 18 /2 ) ^ 2 = 9 ^ 2 = 81 ] to both sides
a ^ 2 - 18 a + 81 = - 10 + 81
a ^ 2 - 18 a + 81 = 71
( a - 9 ) ^ 2 = 71
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Becouse :
( a - 9 ) ^ 2 = a ^ 2 - 2 a * 9 + 9 ^ 2 = a ^ 2 - 18 + 81
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( a - 9 ) ^ 2 = 71 Take square root to both sides
a - 9 = + OR - sqroot ( 71 ) Add 9 to both sides
a - 9 + 9 = 9 + OR - sqroot ( 71 )
a = 9 + OR - sqroot ( 71 )
The solutions are :
a = 9 - sqroot ( 71 )
and
a = 9 + sqroot ( 71 )
Now you have two set of solutions of this problem :
1 )
a = 9 - sqroot ( 71 )
b = 18 - a
b = 18 - [ 9 - sqroot ( 71 ) ]
b = 18 - 9 + sqroot ( 71 )
b = 9 + sqroot ( 71 )
2 )
a = 9 + sqroot ( 71 )
b = 18 - a
b = 18 - [ 9 + sqroot ( 71 ) ]
b = 18 - 9 - sqroot ( 71 )
b = 9 - sqroot ( 71 )
Final solutions :
1 )
a = 9 - sqroot ( 71 )
b = 9 + sqroot ( 71 )
2 )
a = 9 + sqroot ( 71 )
b = 9 - sqroot ( 71 )
You want to find two numbers whose product is 10 and whose sum is 18.
So solution 1 and solution 2 are same solution.
The numbers are:
9 - sqroot ( 71 )
and
9 + sqroot ( 71 )
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To find the two numbers whose product is 10 and sum is 18, we can set up a system of equations:
Let's say the two numbers are x and y.
From the problem, we have two equations:
1) x * y = 10 (product is 10)
2) x + y = 18 (sum is 18)
To solve this system of equations, we can use the substitution method:
Let's solve equation 2 for x:
x = 18 - y
Now we substitute this value of x in equation 1:
(18 - y) * y = 10
Expanding and rearranging:
18y - y^2 = 10
y^2 - 18y + 10 = 0
We now have a quadratic equation. To solve it, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
For our equation:
a = 1, b = -18, c = 10
Plugging these values into the quadratic formula, we get:
y = [18 ± √((-18)^2 - 4(1)(10))] / 2(1)
y = [18 ± √(324 - 40)] / 2
y = [18 ± √284] / 2
y = [18 ± 2√71] / 2
y = 9 ± √71
So, the y values are: 9 + √71 and 9 - √71.
Now, let's substitute these y values back into equation 2 to find the corresponding x values:
When y = 9 + √71,
x + (9 + √71) = 18
x = 18 - 9 - √71
x = 9 - √71
When y = 9 - √71,
x + (9 - √71) = 18
x = 18 - 9 + √71
x = 9 + √71
Therefore, the two numbers are 9 + √71 and 9 - √71, whose product is 10 and sum is 18.