What is th e relation between two quadratic equations if they have a common root?

Question

Please let me know the condition for two quadratic equations for having a common root.....

Answer 1

Let's suppose the two quadratic equations are of the form:
x^2 + ax + b = 0 and
x^2 + cx + d = 0

Also, let's call the common root r.
Now the form of a quadratic equation with two roots r and s is: x^2 -(r+s)x + rs = 0

So if we call the roots of the first equation r and r1 and the roots of the second equation r and r2, then:
b = r*r1, d = r*r2
a = -r - r1 and c = -r - r2

Therefore
r1/r2 =
b/d = (a+r)/(c+r)

and that last line is the condition relating the two quadratic equations.

Answer 2

If two equations represents lines that are parallel, they have same slopes. If two lines are perpendicular, their slopes are reciprocative inverses of each other. But if two quadratic equations have common root, its very difficult to form a relation between the two.
There are many ways in which one equation could be related to other. If you could advise on any particular aspects that you are looking for, it would be much easier to work on solution

Answer 3

Here's a relationship. If you divide one by the other using polynomial long division, the remainder will be a constant multiple of (x-r), where r is the common root.

For instance, let f(x)=x^2-x and g(x)=x^2+x-2.
If you divide g by f you get 1 with remainder 2x-2, a multiple of x-1. It is easy to see that g(1)=0, and that tells you that 1 is a root of f as well.

Answer 4

right this is yet differently to think of approximately it: because of the fact the product is -6, the roots would desire to be: -one million & 6, one million & -6, 2 & -3, -2 & 3 in basic terms the 1st set supplies a sum of 5 for this reason the roots are -one million and 6 So equation has form a(x+one million)(x-6) = 0 ? considering that all equations have coefficient one million for x², then a=one million (x + one million)(x - 6) = 0 x² + x - 6x - 6 = 0 x² - 5x - 6 = 0 answer: b)

Answer 5

let the 2 quadratic equations be

a'(x^2) + b'x + c' =0
a''(x^2) + b''x + c'' =0

then

if they have common roots then relation is

a'/a'' = b'/b'' = c'/c''

Answer 6

a common root would indicate that the graphs of the functions cross at that point.

Answer 7

if an equation has a root saying a , this equation can be divided by (x-a)

So, both equations can be factored with (x-a) as factor

Answer 8

One of their solutions are the same
Please give me best answer thanks!