Incomplete Quadratic Equations

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Source: algebrica.org — CC BY-NC 4.0 https://algebrica.org/quadratic-equations/

Definition

A quadratic equation is considered incomplete when one or both of the terms $bx$ and $c$ are absent from the standard form $ax^2 + bx + c = 0$, provided the term $ax^2$ is present. These equations admit direct solution methods that do not require the quadratic formula or factorization.

When both $b$ and $c$ are equal to zero, the equation reduces to:

ax^2 = 0, \quad a \neq 0

Dividing both sides by $a$, which is nonzero by assumption, gives $x^2 = 0$, and the only real solution is $x = 0$ for every admissible value of $a$.

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Graphically, the equation represents a parabola with its vertex at the origin $(0, 0)$, symmetric about the y-axis. The graph opens upward if $a > 0$ and downward if $a < 0$; the magnitude of $a$ determines the width of the parabola. Although the equation has a single solution, the function has a double root at $x = 0$: the x-axis is tangent to the parabola at the origin.

The case b = 0

When $b = 0$, the equation takes the form:

ax^2 + c = 0, \quad a \neq 0,\, c \neq 0

Isolating $x^2$ gives:

x^2 = -\frac{c}{a}

The equation represents a parabola symmetric about the y-axis. When $a$ and $c$ have opposite signs, the quantity $-c/a$ is positive and the equation has two distinct real solutions:

x_{1,2} = \pm\sqrt{-\frac{c}{a}}

The parabola intersects the x-axis in two points symmetric with respect to the origin. When $a$ and $c$ have the same sign, the quantity $-c/a$ is negative and the equation has no real solutions:

-\frac{c}{a} < 0 \implies x \notin \mathbb{R}

The parabola lies entirely above or below the x-axis and does not intersect it.

The case c = 0

When $c = 0$, the equation takes the form:

ax^2 + bx = 0, \quad a \neq 0,\, b \neq 0

Factoring out $x$ gives $x(ax + b) = 0$. Applying the zero product property, either $x = 0$ or $ax + b = 0$, and the equation has two distinct real solutions:

x_1 = 0 \qquad x_2 = -\frac{b}{a}

Examples

Consider the equation $3x^2 = 0$. Since both $b$ and $c$ are zero, the only solution is $x = 0$.

Consider the equation $2x^2 - 8 = 0$. This is of the form $ax^2 + c = 0$ with $a = 2$ and $c = -8$. Since $a$ and $c$ have opposite signs, two real solutions exist. Isolating $x^2$ gives:

x^2 = \frac{8}{2} = 4

Taking the square root of both sides yields the two solutions:

x_{1,2} = \pm\sqrt{4} = \pm 2

Consider the equation $x^2 + 5 = 0$. Here $a = 1$ and $c = 5$ have the same sign, so $-c/a = -5 < 0$. The equation has no real solutions.

Consider the equation $3x^2 - 6x = 0$. This is of the form $ax^2 + bx = 0$ with $a = 3$ and $b = -6$. Factoring out $x$ gives $x(3x - 6) = 0$, and the two solutions are:

x_1 = 0 \qquad x_2 = \frac{6}{3} = 2

A common error to avoid

For equations of the form $ax^2 + bx = 0$, a frequent error consists in dividing both sides by $x$ when the equation is written as:

ax^2 = bx

Dividing by $x$ is not a valid operation here, since $x = 0$ is itself a solution and division by zero is undefined. This manipulation eliminates the root $x = 0$ and reduces the equation to a linear one, producing only the solution $x = -b/a$. The correct approach is to collect $x$ as a common factor, as shown above.