Okay, converting between quadratic forms. This one's factored form into standard form. Factored form, again, a times x minus p times x minus q, and we're going to convert that into ax squared plus bx plus c of standard form. One thing to notice, again, the a values. The a values are always going to stay the same.
So our a value in this example problem is negative 3 in factored form, so when we're in standard form, Our a value, the number in front, should also be negative 3 again. All right, so let's run through these. If you're going from factored form into standard form, all it is is simplifying.
All you have to do is, it's two quick steps. You're going to FOIL the two binomials and then distribute. So let's do that. Start with FOIL.
So this one I got y equals negative 3 times. x minus 6 times x plus 1 binomial times binomial. So we'll do the first, the outside, the inside, and the last.
And that's going to give us the negative 3. It's going to stay out front for now. x times x, x squared. Outside, x times 1, 1x. Inside, negative 6 times x is negative 6x.
And then last, negative 6 times 1 is negative 6. Now again, part of FOIL, after you do the FOIL, you have to combine like terms if you can. And in this case, we can. Almost always the inside-outside.
Now again, you could skip that step. You could do the FOIL and do a trinomial in your head. That's fine.
But first problem, I'll show all the steps. That negative 3 is just hanging out front. X squared, combine them.
Inside outside terms give you negative 5x 1 1x minus 6x and minus 6 and then your instinct on this Next part is probably correct. You're gonna want to distribute that number in front So that a value that number in front if there is one you're gonna have to distribute through the trinomial. So we distribute the negative 3. Negative 3 times x squared is negative 3x squared. Negative 3 times negative 5x is positive 15x. And then negative 3 times negative 6 is plus 18. And that is standard form.
Now again, it's not asking for this, it's just asking to convert, but when you're doing these, you should go through and just figure out what you could find. So from the original problem, it was in factored form, so you knew the x-intercepts right away, the x-intercepts are the opposite, the p and the q. So the x-intercepts in that one was 6 and negative 1. Now again, standard form, standard form, you could find the y-intercept right away. The y-intercept is always a c-value, so in this... problem the y intercept is 18 so just going through these problems and finding you know the getting the information that you can is always a good kind of a good practice all right let's do one more if you think you're comfortable you can skip it uh let's do uh so we have f of x again f of x and y are interchangeable let's do f of x equals Negative 1 half x plus 4 times x plus 10. Again, two quick steps.
Foil, distribute, and then you're done. So let's foil. The a value is negative 1 half. So again, the a value should stay. negative 1 half in our final answer Foil first outside inside last x squared outside 10x inside 4x it's gonna give me 14x and then last 4 times 10 is 40. All right distribute Distribute the negative 1 half.
Negative 1 half. times x squared negative one half times 14x negative 7x and negative one half times 40 is negative 20. and again that's it now we're in standard form okay again notice the a value the a value doesn't change it never changes in any of these problems negative one half is what the a value that started negative one half is the a value we ended with all right that is factored form into standard form okay that one's pretty easy basically two steps foil combine if you can distribute done