Incomplete quadratic equations

We know that the general form of a quadratic equation is $a x^{2} + b x + c = 0$ . 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 $b x + c = 0$ . Its immediate solution is $x = - \frac{c}{b}$ . 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 $a x^{2} + c = 0$ and we can apply the formula, but it is easier to solve it by isolating the unknown: $x = \pm \sqrt{\frac{- c}{a}}$

Example

$x^{2} - 16 = 0$

$x = \pm \sqrt{\frac{16}{4}} = \pm \sqrt{4} = \pm 2 = {\begin{matrix} x_{1} = 2 \\ x_{2} = - 2 \end{matrix}$

When $c = 0$ the equation is $a x^{2} + b x = 0$ .

In this case we just extract common factor: $x \cdot (a x + b) = 0$ . 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:

$x = 0$

$a x + b = 0 \Rightarrow x = - \frac{b}{a}$ .

Example

$12 x^{2} - 4 x = 0$

$x_{1} = 0 x_{2} = \frac{1}{3}$

Quadratic equations such as:

$a x^{2} + c = 0 a x^{2} + b x = 0$ are called incomplete quadratic equations.