why set quadratic equation to zero?

Question

so i don't really get it what is the exact point of setting it to zero? I mean why can't it be set to 20 or something else. Why is it necessary to have to group all numbers on one side to equal to zero? And on graphs why is a solution have to be x intercepts? why not any other number on a line? And so, then whats the point of a parabola if you are only looking for x intercepts?

please describe in detail. Thank you so much for your time and effort!

Answer 1

so i don't really get it what is the exact point of setting it to zero? I mean why can't it be set to 20 or something else. Why is it necessary to have to group all numbers on one side to equal to zero?



• This is to take advantage of the "zero product property" (which goes under a variety of names). Basically, if a * b = 0, then a or b (or both) has to be 0.



Here's an example with three factors. Let's say that (x - 3)(x + 4)(x - 5) = 0



Then (i) if x - 3 = 0, x = 3

… (ii) if x + 4 = 0, x = -4

… (iii) if x - 5 = 0, x = 5



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And on graphs why is a solution have to be x intercepts? why not any other number on a line? And so, then whats the point of a parabola if you are only looking for x intercepts?



• All of these things are just tools to help you solve a problem. Sometimes you DO want to know various other details of the parabola, not just the x-intercepts.



A typical problem might involve, for example, the area of a rectangular paddock that can be enclosed by 100 ft of fencing. Without going into all the details—I can if you like—you'd end up with area as a function of the length of one side. A graph of the results will be parabolic, and each point on the parabola will represent one possible outcome.



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Answer 2

You set the equation to zero in order to find the x intercepts. You make a valid point with your question about other points. Many times points other than the zeros are significant. But when it comes to graphing the parabola the x intercepts are easier to solve for that any two other points. Combine the x intercepts with the apex and then you have all the data you need to graph the parabola. But that does not mean that the other points don't matter, we are simply searching for the easiest method to get three points including the apex.



In practical applications the zeros are often significant. For example a projectile on earth travels in a parabolic arc. If you set your coordinate system so that the ground as zero then finding the zeros of that parabola will give you information about when the projectile hits the ground. But if you are worried about the projectile reaching a 100 foot cliff, then 100 feet becomes the significant point.

Answer 3

By setting the equation equal to zero and factoring, you can find the values of x which will produce a zero value for y. By convention, those values of x are called the roots (or solutions) of the equation. (Having the equation set equal to zero is also a requirement for using the Quadratic Formula.)



The vertex of a parabola will always be midway between the x-intercepts. Most quadratic solving problems ask for the value(s) of x that produce a zero value for y.



You don't *always* have to set the equation equal to zero. In the case of completing-the-square method of solving quadratics, you arrange and adjust the equation so that the left side factors to a square and has some non-zero number on the right side. You can then take the square root of both sides to solve for x.

Answer 4

Why is it necessary to have to group all numbers on one side to equal to zero?

- So you can apply the quadratic formula



And on graphs why is a solution have to be x intercepts?

- because y=0 at x intercepts



looking at x-intercepts <---> solutions of the quadratic equation

Answer 5

Because you can very easily represent a polynomial by its factors which are determined by its zeros. The reason is the zero product property, which states that if P₁P₂P₃•... = 0, then P_i = 0 for at least one 1 ≤ i. This is only true for 0, not any other constant because 0 is the only real number with no defined multiplicative inverse (there is no number which when multiplied with 0 gives 1). Also, I wrote the property with P, but those don't have to be polynomials, they can be any object which prescribes to the zero product property (there are fields where this property doesn't apply to "0", but to some other element, but now we're getting rather abstract).