Question

Say true or false.
Every quadratic equation has exactly one root.

Answer 1

Hint: We know that if we need to prove something to be true, we have to do it for all the possible things related to it but when we require to prove something as false, we just need to give an example to disprove. We will just give an example here.

Complete step by step answer:
Let us say if possible this statement: “Every quadratic equation has exactly one root” be true.
Then this statement must be true for all the quadratic equations we can possibly form.
Now, we will consider the quadratic equation: \[{x^2} - 1 = 0\].
Now, we have the equation: \[{x^2} - 1 = 0\] with us.
Taking 1 from subtraction in LHS to addition in RHS, we will get:-
\[ \Rightarrow {x^2} = 1 + 0\]
We can rewrite it as follows:-
\[ \Rightarrow {x^2} = 1\]
Now taking square root on both the sides, we will get:-
\[ \Rightarrow x = \sqrt 1 \]
Simplifying the RHS will lead us to:-
\[ \Rightarrow x = \pm 1\]
(Because we know that both ${\left( 1 \right)^2} = 1$ and ${\left( { - 1} \right)^2} = 1$ as well)
Now, we have got two roots that are 1 and -1.
Hence, our assumption was wrong and not every quadratic equation has exactly one root.

Therefore, the given statement is false.

Note:
The students might make the mistake of not considering both the possibilities of getting a positive as well as negative root and thus end up with one root 1 and answer the given question incorrectly. The students must know that even if they consider the equation \[{x^2} = 0\], they still get two roots which are both equal to 0. It is just the case that both the roots are equal to each other but it still has 2 roots.
The students must note that an equation has as much number of roots as its degree always. If you have a linear equation, you will have one root, quadratic – 2, cubic – 3 and so on. It is just not necessary that all roots will be real, some might be imaginary.