LOCATION OF ROOTS :
quadratic equations when the roots are located/ specified on the number line with variety of constraints: Consider `f(x) =ax^2 + bx + c` with `a > 0`.
`text(TYPE-1:)` Both roots of the quadratic equation are greater than a specified number say `(d).` The necessary and sufficient condition for this are :
`(i)` `a > 0 ; \ \ \ \ \ \ (ii) D >= 0;\ \ \ \ \ \ \ \ (iii) f(d) > 0 ; \ \ \ \ \ \ \ \ (iv) - b/2a > d`
`text(Note: )` If `a < 0` then intercept accordingly.
`text(TYPE-2:)` Both roots lie on either side of a fixed number say `(d)`. Alternatively one root is greater than `d` and other less than `d` or `d` lies between the roots of the given equation.
`text(Conditions for this ) \ \ \ \ (i) a > 0` and `(ii) f(d) <0` or
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (i) a > 0` and `(ii) f(d) < 0` Note that no consideration for discriminant will be useful here.
`text(TYPE-3 :)` Exactly one root lies in the interval `( d, e)` when `d < e.`
Conditions for this are:
`(i) a ne0 `
`(ii) f(d) - f(e) < 0`
(iii) An another case arises when ` f(d) - f(e) = 0`
then we have to check end points
For `f(d) = 0; \ \ \ \ \ \ \ \ \ \ ` i.e., one root is `"d"`
Check if other root lies between `"d"` and `"e"` or not. If yes then we will include that point otherwise we will exclude that point
similarly for `f(e) = 0`
we will check for the other root and find out if it lies between `"d"` and `"e"` or not.
`text(Note :) ` lf `f(d) f(e) < 0` then exactly one root lies in the interval `(d, e)` but not the converse.
`text(TYPE-4:)`
When both roots are confined between the number `d` and `e (d < e).` Conditions for this are
`(i) a > 0 ;\ \ \ \ \ \ \ \ (ii) D >= 0 ;\ \ \ \ \ \ \ \ (iii) f(d) > 0 ;\ \ \ \ \ \ \ \ (iv) f(e) > 0`
`text(TYPE-5:)`
One root is greater thane and the other root is less than d.
Conditions are:
`(i) f(d) < 0 and f(e) < 0` if `(a > 0)`
`text(TYPE-1:)` Both roots of the quadratic equation are greater than a specified number say `(d).` The necessary and sufficient condition for this are :
`(i)` `a > 0 ; \ \ \ \ \ \ (ii) D >= 0;\ \ \ \ \ \ \ \ (iii) f(d) > 0 ; \ \ \ \ \ \ \ \ (iv) - b/2a > d`
`text(Note: )` If `a < 0` then intercept accordingly.
`text(TYPE-2:)` Both roots lie on either side of a fixed number say `(d)`. Alternatively one root is greater than `d` and other less than `d` or `d` lies between the roots of the given equation.
`text(Conditions for this ) \ \ \ \ (i) a > 0` and `(ii) f(d) <0` or
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (i) a > 0` and `(ii) f(d) < 0` Note that no consideration for discriminant will be useful here.
`text(TYPE-3 :)` Exactly one root lies in the interval `( d, e)` when `d < e.`
Conditions for this are:
`(i) a ne0 `
`(ii) f(d) - f(e) < 0`
(iii) An another case arises when ` f(d) - f(e) = 0`
then we have to check end points
For `f(d) = 0; \ \ \ \ \ \ \ \ \ \ ` i.e., one root is `"d"`
Check if other root lies between `"d"` and `"e"` or not. If yes then we will include that point otherwise we will exclude that point
similarly for `f(e) = 0`
we will check for the other root and find out if it lies between `"d"` and `"e"` or not.
`text(Note :) ` lf `f(d) f(e) < 0` then exactly one root lies in the interval `(d, e)` but not the converse.
`text(TYPE-4:)`
When both roots are confined between the number `d` and `e (d < e).` Conditions for this are
`(i) a > 0 ;\ \ \ \ \ \ \ \ (ii) D >= 0 ;\ \ \ \ \ \ \ \ (iii) f(d) > 0 ;\ \ \ \ \ \ \ \ (iv) f(e) > 0`
`text(TYPE-5:)`
One root is greater thane and the other root is less than d.
Conditions are:
`(i) f(d) < 0 and f(e) < 0` if `(a > 0)`