Incomplete quadratic equations

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.