Nature of roots of a quadratic equation `ax^2 + bx + c = 0` implies whether the roots are real or imaginary by analyzing the quantity D.
1. Let `a,b, c in R` and `a!=0` , then the equation `ax^2 + bx + c = 0` has
(i) real and distinct roots if and only if `D > 0.`
(ii) real and equal roots if and only if `D = 0.`
(iii) complex roots with non-zero imaginary parts if and only if `D < 0.` If `p + iq` (where `p, q in R, q != 0`) is one root of `ax^2 + bx + c = 0`, then second roots will be `(p - iq).`
2. If `a, b, c in Q` and D is a perfect square, then `ax^2 + bx + c = 0` has rational roots.
3. If `a, b, c in Q ` and `p + sqrtq` ( where, `p, q in Q`) is an irrational root of `ax^2 + bx + c = 0`, then other root will be `(p - sqrtq )`.
4. If `a = 1, b, c in I` are roots of `ax^2 + bx + c = 0`, are rational numbers, then these roots must he integers
5. If `ax^2+ bx + c = 0` is satisfied by more than two distinct complex numbers, then it becomes an identity i.e. `a = b = c = 0`.
6. If the roots of `ax^2 + bx + c = 0` are both positive, then the signs of `a` and `c` should be like and opposite to the sign of `b.`
7. If the roots of `ax^2 + bx + c = 0` are both negative, then signs of `a ,b` and `c` should be like.
8. If the roots of `ax^2 + bx + c = 0` are equal in magnitude but opposite in sign, then `b = 0` and `c < 0.`
9. If the roots of `ax^2 + bx + c = 0` are reciprocal to each other, then `c = a.`
10. In the equation `ax^2 + bx +c = 0` (where, `a,b,c in R`), if `a + b + c = 0` , then roots are `1, c/a` and if `a - b +c = 0` then the roots are `-1` and `-c/a`
Nature of roots of a quadratic equation `ax^2 + bx + c = 0` implies whether the roots are real or imaginary by analyzing the quantity D.
1. Let `a,b, c in R` and `a!=0` , then the equation `ax^2 + bx + c = 0` has
(i) real and distinct roots if and only if `D > 0.`
(ii) real and equal roots if and only if `D = 0.`
(iii) complex roots with non-zero imaginary parts if and only if `D < 0.` If `p + iq` (where `p, q in R, q != 0`) is one root of `ax^2 + bx + c = 0`, then second roots will be `(p - iq).`
2. If `a, b, c in Q` and D is a perfect square, then `ax^2 + bx + c = 0` has rational roots.
3. If `a, b, c in Q ` and `p + sqrtq` ( where, `p, q in Q`) is an irrational root of `ax^2 + bx + c = 0`, then other root will be `(p - sqrtq )`.
4. If `a = 1, b, c in I` are roots of `ax^2 + bx + c = 0`, are rational numbers, then these roots must he integers
5. If `ax^2+ bx + c = 0` is satisfied by more than two distinct complex numbers, then it becomes an identity i.e. `a = b = c = 0`.
6. If the roots of `ax^2 + bx + c = 0` are both positive, then the signs of `a` and `c` should be like and opposite to the sign of `b.`
7. If the roots of `ax^2 + bx + c = 0` are both negative, then signs of `a ,b` and `c` should be like.
8. If the roots of `ax^2 + bx + c = 0` are equal in magnitude but opposite in sign, then `b = 0` and `c < 0.`
9. If the roots of `ax^2 + bx + c = 0` are reciprocal to each other, then `c = a.`
10. In the equation `ax^2 + bx +c = 0` (where, `a,b,c in R`), if `a + b + c = 0` , then roots are `1, c/a` and if `a - b +c = 0` then the roots are `-1` and `-c/a`