Question

If the signs of a and c are opposite and b is real, the roots of the quadratic equation $a{x^2} + bx + c = 0$ are?
a) real and distinct
b) real and equal
c) imaginary
d) both roots are positive

Answer 1

we can know the location of roots of quadratic equations, the most important things are discriminant. For example, if the discriminant is more than $0$,it has real and distinct roots. The discriminant of ${x^2}$ and c are also very important to determine whether the graph is opening on positive y axis or negative y.

Complete step-by-step solution:
We will find the discriminant D of a quadratic equation $a{x^2} + bx + c = 0$ .
$D = {b^2} - 4ac$
We were given in question, a and c are of opposite signs.
So, for$\;1st$case :
We assume a greater than and c less than 0. So,
$a > 0$and $c < 0$,
$ac < 0$
So, $- ac > 0$
we have discriminate D,
$D = {b^2} - 4ac$
$= {b^2} + 4( - ac)$
$= {b^2} + 4( - ac) > 0$
$= D > 0$
We have a discriminant greater than 0.
In case $2$:
We assume c greater than and a less than 0.
a<0 and c>0
$ac < 0$
So,$- ac > 0$
Similarly
$= D > 0$
Since, the discriminant is always greater than 0.
The quadratic equation$a{x^2} + bx + c = 0$ has roots are real and distinct.

Note: Quadratic equations are the polynomial equations of degree 2 in one variable of type $f(x) = a{x^2} + bx + c$where a, b, c, ∈ R and a is not equal to$0$. It is the general form of a quadratic equation where ‘a’ is called the leading coefficient and ‘c’ is called the absolute term of f (x).