Question 986986: a) Is y = x^2 + 7x + 12 a quadratic equation*? Why or why not?
b) Can it be solved by factoring? If yes, solve it. If not, why not?
c) As it happens, the equation y = x^2 + 3x - 23 also cannot be solved by factoring. One method of solving quadratic equations, however, that will solve ALL of them, is called using the quadratic formula. Can you state and explain the quadratic formula?
d) Use the quadratic formula to solve y = x^2 + 3x - 23 and explain your steps.
Thanks
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
A quadratic equation is any equation of the form  where  . For your purposes at this point in your study of mathematics, assume that all of the coefficients are real numbers. However,  is a quadratic equation if and only if  is equal to some specific value such that with appropriate algebraic manipulation the equation can be arranged in to the form. Without such a restriction on the value of , is a quadratic function.
Quadratic functions cannot be "solved" in the commonly accepted sense of finding specific numbers that make the equation a true statement. The only thing you can say about the solution set of is that it is \ \in\ \mathbb{R}^2\,:\,y\ =\ ax^2\ +\ bx\ +\ c\right\}) . Trivial in the extreme. It is only when you specify a particular value for that the function becomes a quadratic equation that is solvable for a particular pair of values of 
In fact, all quadratic equations can be reduced to the product of a pair of factors, although they may not necessarily be rational or even real. However, unless the factors are rational, finding the factors by ordinary factorization means may be insurmountably challenging. All of this to say that you stated this question incorrectly. What you are really being asked is if the quadratic equation in question is factorable over the rationals. Fortunately, there is a simple way to tell: The quadratic is factorable over the rationals if and only if  is a perfect square. In the case of your example,  and  is a perfect square. Therefore the quadratic trinomial  is factorable over the rationals. Therefore,  can be solved by factoring. Hint: 3 + 4 = 7 and 3 * 4 = 12.
The quadratic formula is:
Plug in your numbers and do the arithmetic.
John
My calculator said it, I believe it, that settles it
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