Question

Graph of a quadratic equation is always a
(a) straight line
(b) circle
(c) parabola
(d) hyperbola

Answer 1

Hint: We will look at the general quadratic equation which is $a{{x}^{2}}+bx+c=0$. We will use the discriminant of the quadratic equation and its relation with the roots of the quadratic equation and try to interpret the geometric meaning. We will use this information to deduce the required answer.

Complete step by step answer:
The general quadratic equation is given as $a{{x}^{2}}+bx+c=0$ where $\text{a, b, c}$ are real numbers. The discriminant of the quadratic equation is given by $D={{b}^{2}}-4ac$. We know that
(i) if $D > 0$, then the quadratic equation has two distinct, real roots;
(ii) if $D=0$, then the two roots are equal and real; and
(iii) if $D < 0$, then the quadratic equation has roots that are not real.
Now, we know that the roots of an equation are the intercepts on x axis and the roots of the equation are the solution of $a{{x}^{2}}+bx+c=0$, hence, the y coordinate is $0$.
So, if the quadratic equation has two distinct real roots, then they are two points on the x axis. This means that the graph of the quadratic equation cuts the x axis at two points.
If the roots of the quadratic equation are equal and real, then this means that the x axis is tangent to the graph of the quadratic equation and touches it at exactly one point.
If the quadratic equation has no real roots, then the graph of that equation does not intersect the x axis at any point.
Therefore, the graph of a quadratic equation is always a parabola.

So, the correct answer is “Option C”.

Note: We can eliminate two of the options right away by looking at the different possibilities of roots that a quadratic equation can have. The straight line will never have two distinct real roots, so we eliminate option (a). The hyperbola will not have two equal roots, hence we eliminate option (d).