Wilma is thinking of 2 numbers. The sum is 2 and the product is -24. Use a quadratic equation to find the two numbers.

This is an Algebra II word problem.

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You should use the Lagrange's resolvents to find the solutions to the quadrativ equation `x^2 - px + q = 0` .

You should know that p represents the sum of the numbers and p represents the product of the two numbers.

Since the problem provides the values of the sum and product, you may substitute the values for p and q such that:

`x^2 - x+ (-24) = 0`

`x^2 - 2x- 24 = 0`

Notice that the solutions to the quadratic equation above represent the two missing numbers.

You may solve the quadratic equation using the following quadratic formula such that:

`x_(1,2) = (-b+-sqrt(b^2-4ac))/(2a)`

Identifying the coefficients a,b,c yields:

`a = 1 , b = -2 , c = -24`

`x_(1,2) = (2+-sqrt(4 + 96))/2 => x_(1,2) = (2+sqrt100)/2`

`x_(1,2) = (2+-10)/2 => x_1 = (2+10)/2 = 6 ; x_2 = (2-10)/2 = -4`

**Hence, evaluating the two missing numbers, under the given conditions, yields `x_1 = 6` , `x_2 = -4` .**

Let the first number be x

then the other number = 2-x

the product of the numbers is x(x-2) = -24

=> x^2-2x = -24

=> x^2-2x+24 = 0

x = [-(-2)+sqrt[(-2)^2-4*1*(-24)]/(2*1) and

x = [-(-2)-sqrt{(-2)^2-(4*1*(-24)}]/(2*1)

=> x = [2+sqrt(100)]/2, [2-sqrt(100)]/2

=> x = 6, -4

The correspnonding other number is x-2 = 4, -6

But sum is 2 only for the numbers 6 and -4 so

**The numbers are 6 and -4 for which the sum is 2 and product is -24**

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