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Write whether the following statements are true or false. Justify your answers.
If the coefficient of $ x^{2} $ and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
Given:
If the coefficient of \( x^{2} \) and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
To do:
We have to find whether the given statement is true or false.
Solution:
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.
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