Question

The product of two numbbers whose sum is 23 [define a variable and write quadratic expression]

Answer 1

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Step-by-step explanation:

Let x = the first number

Let y = the second number

Using the concept and variables, we set up equation based on the description.

x + y = 23 eq1

xy = Product eq2

Substitute eq1 into eq2. Lets get eq2 in terms of x. This will allow us to get a function.

Product = x(23 - x)

Product = -x2 + 23x

We have a quadratic function. When the coefficient of a quadratic function's leading term is negative, it indicates that we will have a maximum. To find the largest product possible, we need to find the maximum value of this function. We do this by finding the vertex.

Vertex has coordinate (h, k).

h = -b / 2a

where:

a = -1

b = 23

Plug in these values to find h. This will be x coordinate of the maximum value.

h = -23 / (2*-1)

h = -23 / (-2)

h = 11.5

x = 11.5

Substitute this value of x into eq1 to find y.

y = 23 - x

y = 23 - 11.5

y = 11.5