Write whether the following statements are true or false. Justify your answers.
Every quadratic equation has exactly one root.

To do:

We have to find whether the given statements are true or false.

Solution:

(i) We know that,

A quadratic equation has two roots.

For example,

$x^2-16=0$ has two distinct roots $-4$ and $4$.

Hence, the given statement is false.

(ii) We know that,

A quadratic equation has two roots(real or imaginary).

For example,

$x^2+9=0$ has no real roots.

Hence, the given statement is false. 

(iii) We know that,

A quadratic equation has two and only two roots(real or imaginary).

Hence, the given statement is false. 

(iv) We know that,

A quadratic equation has two roots(real or imaginary).

Hence, the given statement is true. 

(v) In a quadratic equation $ax^2+bx+c = 0$, if $a$ and $c$ have opposite signs, then $ac<0$

This implies,

$D=b^2 – 4ac > 0$

The discriminant is always positive, so it always has real roots.

Hence, the given statement is true. 

(vi) In a quadratic equation $ax^2+bx+c = 0$, if $a$ and $c$ have same signs, then $ac>0$

This implies,

$D=(0)^2 - 4ac$

$=-4ac<0$

The discriminant is always negative, so it has no real roots.

Hence, the given statement is true.  

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Updated on: 10-Oct-2022

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