Incomplete Quadratic Equations
What are incomplete quadratic equations
A quadratic equation is considered incomplete if it lacks one of the terms from the standard form $a x 2 + b x + c = 0$, 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: \(a x^{2} = 0 a \neq 0\) In this case, the equation has one real solution $x = 0 \forall a \neq 0$.
Graphically, the equation represents a parabola with its vertex at the origin $( 0 , 0 )$. It touches the x-axis but does not cross it. The graph opens upward if $a > 0$, and downward if $a < 0$. The value of $a$ also controls how wide or narrow the parabola appears.
Although the equation has only one solution, it remains a quadratic equation: the function has a double root at zero, meaning the x-axis is a tangent to the parabola at the origin.
If $b$, the coefficient of the linear term $x$ is equal to zero, we have: \(a x^{2} + c = 0 a \neq 0 , c \neq 0\)
In this case the solution is: \(x^{2} = - \frac{c}{a}\)
In this case, the equation represents a parabola with no linear term, which means it is symmetric about the y-axis. If $\frac{c}{a} < 0$, the graph intersects the x-axis in two symmetric points. If $\frac{c}{a} > 0$, there are no real solutions and the parabola does not touch the x-axis.
If $a$, the coefficient of the quadratic term $x^{2}$ and the constant $c$ have different signs, the equation has two distinct real solutions:
\[x_{1 , 2} = \pm \sqrt{- \frac{c}{a}}\]This means the parabola is symmetric about the y-axis and intersects the x-axis in two distinct points. These points are symmetric with respect to the origin.
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. \(- \frac{c}{a} < 0 \rightarrow ∄ x \in \mathbb{R}\)
If the constant term $c$ is equal to zero, we have: \(a x^{2} + b x = 0 a \neq 0 , b \neq 0\) Factoring out the common factor we have: $x ( a x + b ) = 0$, and applying the zero product property we obtain: $x = 0 ( a x + b ) = 0$ The equation has two distinct real solutions: \(x_{1} = 0 x_{2} = - \frac{b}{a}\)
We now present a common mistake that often occurs when first approaching quadratic equations.
For equations of the form $a x^{2} + b x = 0$, one must avoid the common error arising from the simplification of the unknown $x$ when the equation, for example, presents itself as:
\[a x^{2} = b x\]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.