Question

Please only answer if you have a serious answer not just “do your work” I’m in 18 classes and my school won’t let me out of any of them. I’m just really confused so any help would be nice!
Write your own quadratic equation in standard form. Convert this quadratic to factored form. Be sure to show all of your work. Post to the discussion board these two equations and the work you did to convert from quadratic form to standard form.
Now write a new, different quadratic function, f of x, in standard form. Convert this quadratic function to factored form. Post to the discussion board both of these functions and the work you completed to convert from the quadratic form of the function to the factored form of the function.
Use your factored equation and your factored function to explain the differences between roots, x-intercepts, zeros, and solutions.

Answer 1

9514 1404 393

Explanation:

1) 2x^2 -5x -3 = 0 . . . . standard form equation

To convert this to factored form, you can look for factors of the product (2)(-3) that have a sum of -5. It can help to start by listing the ways that -6 can be factored. Since we want the sum of factors to be negative, we want to have larger negative factors.

  -6 = (1)(-6) = (2)(-3)

The sums of these factor pairs are -5 (what we want) and -1 (not relevant). We can call these factors p=1 and q=-6.

If a = 2 is the leading coefficient of our standard form quadratic, we want to use these factors in the form ...

  (ax +p)(ax +q)/a . . . . . factored form of the quadratic

  (2x +1)(2x +(-6))/2 . . . .fill in the values we know

  (2x +1)(x -3) . . . . . . .  factor 2 from the second binomial

So, the factored form of the quadratic equation is ...

  (2x +1)(x -3) = 0 . . . . factored form equation

__

2) f(x) = x^2 +7x +10 . . . . standard form quadratic function

Using the thinking process described above, we are looking for factors of 10 that have a sum of 7. We know those are 2 and 5. So, the factored form of the function is ...

  f(x) = (x +2)(x +5) . . . . . . factored form quadratic function

The leading coefficient is 1, so we have no further work to do.

Roots, x-intercepts, zeros

The graph attached below shows this function crosses the x-axis when x=-2 and x = -5. These values of x are variously called "roots", "x-intercepts", and "zeros" of the function. They are values for which the factors and the function are zero. (x+2=0 when x=-2, for example)

Solutions

Often, we are interested in solving the equation ...

  f(x) = 0

For that equation, the solutions are the zeros or x-intercepts or roots. The graph attached also shows solutions for ...

  f(x) = 4

Those solutions are x = -6 and x = -1. The function value is not zero for these values of x, so the roots, x-intercepts, or zeros are not solutions to this equation.