Question

Can anyone tell me how to find range of quadratic equations.

Answer 1

If there is a quadratic equation say ax^2 + bx + c = f(x), then I can write it as:

Answer 2

\[\large{ax^2 + bx + c = 0}\]

\[\large{\implies x^2 + \cfrac{b}{a} + \cfrac{c}{a} = 0}\]

\[\large{\implies (x + \cfrac{b}{2a})^2 + \cfrac{c}{a} - \cfrac{b^2}{4a^2} = 0}\]

Answer 3

Now for all real values,

\[\large{(x+\cfrac{b}{2a})^2 \ge 0}\]

Answer 4

So minimum value of f(x) will occur when \(\large{(x+\cfrac{b}{2a}) = 0}\)

Answer 5

And that minimum value would be:

\[\large{f(x)_{min} = f(\cfrac{-b}{2a}) = \cfrac{c}{a}-\cfrac{b^2}{4a^2}}\]

Answer 6

There is no global maximum of this f(x).

Answer 7

oh can we take an example?

Answer 8

Thus I can write:

\[\large{ax^2 + bx + c \in [\cfrac{c}{a} - \cfrac{b^2}{4a^2}, \infty)}\]

Answer 9

Yeah sure for example take a quadratic equation:

\[\large{f(x) = 2x^2 - 3x + 4}\]

Answer 10

wait i will give

Answer 11

Sure :)

Answer 12

\[\huge \frac{ x ^{2}-x+1 }{ x ^{2}+x+1 }\]

Answer 13

Great :) Couldn't find anything easier ? ;)

Answer 14

This necessarily does not requires to be solved in the above way :)

Answer 15

let's do difficult one's

Answer 16

Okay lets see:

\[\large{\cfrac{x^2 - x + 1}{x^2+x+1} = \cfrac{f(x)}{g(x)}}\]

Answer 17

Now one method is to use maxima and minima

Answer 18

No not calculus method

Answer 19

If a < 0 then the extreme value \(\cfrac{c}{a}-\cfrac{b^2}{4a^2}\) will be a global maximum because the parabola opens downwards.

Answer 20

\[\large y = \frac{ x ^{2}-x+1 }{ x ^{2}+x+1 } \]

solve \(x\)
range of y = domain of x

Answer 21

(y-1)x^2+(y+1)x+y-1 =0

Answer 22

y exists in the range precicely when it makes x, a real number (because, implicitly we're talking about domain and range in real numbers)

Answer 23

when do you get x as a real solution for a quadratic : \(\large ax^2+bx+c=0\) ?

Answer 24

d greater than or = 0

Answer 25

exactly ! set your D >= 0

Answer 26

you will get a quadratic inequality whicih you can solve for the range