A quadratic equation is considered incomplete if it lacks one of the terms from the standard form \(ax² + bx + c = 0\), as long as the \(x^2\) term is present. These equations are easy to solve, and there is no need to use the quadratic formula or the factorization method to find their roots.

  • If \(b\), the coefficient of the linear term \(x\), and the constant \(c\) are equal to zero, we have: \[ax^2 = 0 \quad a \neq 0 \] In this case, the equation has one real solution \(x = 0 \quad \forall a \neq 0\).

  • If \(b\), the coefficient of the linear term \(x\) is equal to zero, we have: \[ax^2 + c = 0 \quad a \neq 0, c \neq 0 \] \[ x^2 = -\frac{c}{a} \]

  • If \(a \), the coefficient of the quadratic term \(x^2\) and the constant \(c\) have different signs, the equation has two distinct real solutions: \[x_{1,2} = \pm \sqrt{-\frac{c}{a}} \]

  • If \(a \), the coefficient of the quadratic term \(x^2\) and the constant \(c\) have the same sign, the value inside the root is negative. In this case the equation has no real solutions. \[-\frac{c}{a} \lt 0 \to \nexists \hspace{10px} x \in \mathbb{R}\]

  • If the constant term \(c\) is equal to zero, we have: \[ax^2 + bx = 0 \quad a \neq 0, b \neq 0\] Factoring out the common factor we have: \[x(ax+ b) = 0\] Applying the zero product property we have: \[x = 0 \quad (ax+ b) = 0\] The equation has two distinct real solutions: \[x_1 = 0 \quad x_2 = -\frac{b}{a}\]


A common error

For equations of the form \( ax^2 + bx = 0\), one must avoid the common error arising from the simplification of the unknown \(x\) when the equation, for example, presents itself as:

\[ax^2 = bx\]

If you lack experience in solving equations, it can be tempting to simplify both sides of the equation incorrectly. This can cause you to lose one or both solutions and convert a quadratic equation into a linear equation. Therefore, it is essential to understand the equation fully and avoid any hasty simplifications.