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

Hint: 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).