A quadratic equation/function can have only two solutions, which are called the "roots" of the quadratic. It cannot have more than two roots, and it can also not have lesser than two roots (though there can be two equal roots).
We can prove the above fact by the following mathematical method:
Therefore our assumption that a quadratic expression ax^2 + bx + c can have three roots 'α', 'ß and 'Γ' is false.
Therefore a quadratic can not have more or lesser than two roots, or, in other words, a quadratic equation cannot have three or more roots
We can prove the above fact by the following mathematical method:
Proof:
| Statement | Reason |
|---|---|
| 1) Let a quadratic expression ax^2 + bx + c have three roots 'α', 'ß' and 'Γ'. | Assumption |
| 2) Therefore (x - α), (x - ß) and (x - Γ) are all factors of the above quadratic. | By the Factor Theorem |
| 3) It means that (x - α)(x - ß)(x - Γ) is a factor of the quadratic. | Product of factors is a factor itself |
| 4) The degree of the above expression is 3 | Because on expanding (x - α)(x - ß)(x - Γ) we get a polynomial in the degree of 3. |
| 5) Therefore, by our assumption, a higher degree polynomial expression is a factor of a lower degree polynomial expression, which cannot be true. | Since the degree of a quadratic expression is 2, which is lesser than 3. |
Therefore our assumption that a quadratic expression ax^2 + bx + c can have three roots 'α', 'ß and 'Γ' is false.
Therefore a quadratic can not have more or lesser than two roots, or, in other words, a quadratic equation cannot have three or more roots
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