# Incomplete Quadratic Equations

*Ref.*quadratic equations

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.