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.