What's the shape of ax^2+bx+c=0 equation?
1 Answer
Note that $ax^2+bx+c=0$ and $y=ax^2+bx+c$ are completely different objects. The former is an equation and the latter is a function.
The quadratic equation $ax^2+bx+c=0$ has two solutions
\[x=\frac{-b+\sqrt{b^2-4ac}}{2a}, x=\frac{-b-\sqrt{b^2-4ac}}{2a} \]
provided $b^2-4ac>0$. Note that these are equations of two veritical lines. If the equation has only one solution ($b^2-4ac=0$), then $ax^2+bx+c=0$ would represent only one straight line. That's why you are getting two straight lines.
However, if you mean $y=ax^2+bx+c$, then it represents the graph of a parabola.
The quadratic equation $ax^2+bx+c=0$ has two solutions
\[x=\frac{-b+\sqrt{b^2-4ac}}{2a}, x=\frac{-b-\sqrt{b^2-4ac}}{2a} \]
provided $b^2-4ac>0$. Note that these are equations of two veritical lines. If the equation has only one solution ($b^2-4ac=0$), then $ax^2+bx+c=0$ would represent only one straight line. That's why you are getting two straight lines.
However, if you mean $y=ax^2+bx+c$, then it represents the graph of a parabola.
Paul F
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Thank you for your explanation
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