Find roots of quadratic equations using sum and product rules for roots.?

Question

I don't understand the sum and product rule can someone please examplain them please!

Answer 1

Let the quadratic equation be ax^2+bx+c=0.
Let the roots be m and n.
Then m+n = -b/a and mn = c/a are the sum and product rules.
So let s = m+n = -b/a and p = mn = c/a.

Then the original equation is equivalent to x^2-sx+p=0.
So given s = sum of roots and p = product of roots,
the two solutions of the original equation are

x = (s +- sqrt(s^2-4p))/2.

This formula is a bit simpler to compute
than the original quadratic equation formula.

Answer 2

The rule says that take any degree polynommial. Put it in descending order and divide by the leading coefficient so that the leading coefficient is one. Then the second coeffecient will be the negative of the sum of ALL the roots of the polynomial and the last term (which will be a constant) will be the PRODUCT of all of the roots.

What does this say in English?

This says that if my polynomial is
f(x)=2x^5+4x^4+6x^3+8x^2+10x+12=0

Divide by two first because I want the leading coefficient to be one.
x^5+2x^4+3x^3+4x^2+5x+6=0


then whatever those roots are (and there are 5 of them by the fundamental theorem of algebra. They can be complex roots involving imaginary terms, they can be irrational, and they can repeat as well). The sum of all the roots is -2 which is the second coefficient. When I multiply all of the roots together, I will get 6 which is the constant at the end.

A Simple Example:
x^2-2x+1=0

Let the roots be a and b. I know from looking at that equation that
a+b=2 and
(a)(b)=1

a=1, b=1 is the solution therefore the two roots are x=1 as a double root.

This is a very bad way of calculating roots, very inefficient.

Answer 3

I think you are talking about the following:

take the quadratic ax^2+bx+c=0, call the roots p and q.

The sum p+q = -b
The product pq=c

It is generally not efficient to use these concepts to solve for roots, but it depends on the information you were given.for example, if you are askedto find the sum of the roots, just give -b as your answer: you don't have to actually calculate the roots.

Answer 4

Sum: r? + r? = –b/a Product: r?r? = c/a a million.) x² – 12x + 22 = 0 r? + r? = –(–12)/a million = 12 r?r? = 22/a million = 22 2.) x² – 22 = 0 r? + r? = 0/a million = 0 r?r? = –22/a million = –22 3.) 2x² – a million/4x = – 8/15 2x² – a million/4x + 8/15 = 0 r? + r? = (–a million/4)/2 = –a million/8 r?r? = (8/15)/2 = 4/15

Answer 5

Use the Mean Value Theorem!