Is $ax^2 + bx = 0$ considered a quadratic equation? Or is it linear, since it simplifies to $ax+b=0$?

Question

I know that a quadratic equation can be represented in the form $$ax^2 + bx + c = 0$$ where $a$ is not equal to $0$, and $a$, $b$, and $c$ are real numbers. However, if there is an equation in the form $$ax^2 + bx = 0$$ would it be classified as a quadratic equation since the conditions are satisfied, or would it be a linear equation since it can be simplified into $ax + b = 0$?

Answer 1

It is a quadratic equation as it satisfies the definition.

Notice that $ax^2+bx=0$ and $ax+b=0$ are not equivalent, the first one has $0$ as a solution for sure and $\frac{-b}a$ as a root as well.

Answer 2

You need only $a≠0$ for $$ax^2+bx+c=0.$$

If the degree of your polynomial is equal to $2$, then you have a quadratic polynomial. This means , your equation $ax^2+bx=0$ is still a quadratic equation, if $a≠0.$

But, the degree of the polynomial $ax+b$ is equal to $1$. This implies, $ax+b=0$ is not a quadratic.

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Small Supplement:

$ax^2+bx=0$ is not equivalent to $ax+b=0$. Because, $x=0$ is not always a root of $ax+b=0.$

Answer 3

A quadratic equation is an equation that can be rearranged as $ax^2+bx+c=0$ where $a$ is not equal to $0$ and $b$ and $c$ are real numbers. If $a=0$ then the equation is linear not quadratic since the $x^2$ has no influence .

Answer 4

Hints

1. If you draw the graph of $y=ax^2+bx$ what shape is it? (Plug in some non-zero values for $a$ and $b$)

2. If you factorize $ax^2+bx=0$ and then apply null factor law, how many solutions are there?