We know that the general form of a quadratic equation is $$ax^2+bx+c=0$$. When some of the coefficients $$a, b$$ or $$c$$ is zero, the solutions can be found in a very simple way.
- If $$a = 0$$, the equation is written as $$bx + c = 0$$. Its immediate solution is $$\displaystyle x=-\frac{c}{b}$$. We will not consider this case since this is not a quadratic equation, but a linear equation or a first degree equation (the greatest exponent of $$x$$ is $$1$$).
- If $$b = 0$$ the equation can be written as $$ax^2+c=0$$ and we can apply the formula, but it is easier to solve it by isolating the unknown: $$x=\pm \displaystyle \sqrt{\frac{-c}{a}}$$
$$x^2-16=0$$
$$$\displaystyle x=\pm \sqrt{\frac{16}{4}}=\pm \sqrt{4}=\pm 2 =\left\{\begin{matrix} x_1=2 \\ x_2=-2\end{matrix}\right.$$$
- When $$c = 0$$ the equation is $$ax^2+bx=0$$.
In this case we just extract common factor: $$x\cdot (ax + b) = 0$$. When the product of two factors is zero, at least one of them must be a zero, so we can obtain the solutions by making each of the factors zero:
$$$x = 0$$$
$$$ax + b = 0 \Rightarrow \displaystyle x= -\frac{b}{a}$$$.
$$12x^2-4x=0$$
$$$\displaystyle x_1=0 \\ x_2=\dfrac{1}{3}$$$
Quadratic equations such as:
$$$ax^2+c=0 \\ ax^2+bx=0$$$ are called incomplete quadratic equations.