Quadratics!!

From Math Wiki

Challenge Problem

I'm thinking of two numbers. Their product is 414 and their sum is 41. What are the two numbers? (Show how this problem looks as a quadratic and show work here.)

Factored to Standard Form

This is how to do a quadratic function in FACTORED form.

starting problem:

                    g(x)= (x+5)(x-2)

you want to multiply the first problem in parenthesis: (x+5), by distributing it into the second set of parenthesis: (x-2).

x(x-2) is the first part of the equation when distributing. Then you have to add 5(x-2) because you're multiplying the 5 from the first part by the second set of parenthesis.

At this point you have:

                       g(x)= x(x-2)+5(x-2)

                       g(x)= x^2-2x+5x-10

This is problem isn't finished yet. You must combine like terms for the final solution.

                      -2x+5x= 3x
    

Therefore: g(x)= x^2+3x-10

There's your answer, I hope it helps! :)

Quadratic Formula

Now let's go ahead and solve the equation.

The equation is:

                 g(x) = ax^2 + bx + c   

The solution is:

                  x = ((-b)+or-(b^2-4*a*c)^1/2)/2*a 

The numbers are: a=1, b=3, c=-10

Since the solution is + or - , there are 2 possible answers.

For the plus answer: x = ((-3) + (3^2 - 4*1*-10)^1/2) / 2*1

                      x = ((-3) + (9- (-40))^1/2) / 2

                      x = ((-3) + (7)) / 2

                      x = 4 / 2

                      x = 2

For the minus answer: x = ((-3) - (3^2 - 4*1*-10)^1/2) / 2*1

                      x = ((-3) - (9- (-40)^1/2) / 2

                      x = ((-3) - (7)) / 2

                      x = -10 / 2

                      x = -5

The solution to the quadratic equation is: x = 2 , -5