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

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 $$ax + b = 0$$?

• We define a polynomial by the highest power present Apr 19, 2021 at 8:30
• It can't be simplified into $ax+b=0$, which has one solution. It can be factorised, like other quadratics, as $x(ax+b)=0$; and this gives two solutions, in the normal way. Apr 19, 2021 at 8:50

## 4 Answers

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.

• Oh I see. Thanks! Apr 19, 2021 at 8:31

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 .

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?

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.

Small Supplement:

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

• I see. But upon multiplying x to both sides of ax + b = 0, we get ax^2 + bx = 0. Would this be a quadratic equation (since a is not equal to 0)? Or would it be linear since it is the same as ax + b = 0? Apr 19, 2021 at 8:30
• @Twilight I added a small supplement. Apr 19, 2021 at 8:41
• @Twilight: Note that you cannot multiply by something on both sides of an equation until you're sure that that something is not zero. Apr 19, 2021 at 8:44