Quadratic Equations
A complete treatment: from definition through every solving method to applications, parameter analysis, and reducible equations.
1. Definition and Standard Form
The General Quadratic Equation
A quadratic equation in one variable $x$ is any equation that can be written in the standard form:
The coefficients carry names you should use fluently:
- $a$ — the leading (quadratic) coefficient. It must be nonzero; otherwise the $x^2$ term vanishes and the equation degenerates to the linear case $bx + c = 0$, which you already know how to solve (callback to Linear Equations).
- $b$ — the linear coefficient.
- $c$ — the constant term (or free term).
Incomplete Quadratics
When one of the non-leading coefficients is zero, the equation is called incomplete. These cases have their own faster routes — reaching for the general formula on them is the fluency failure to prevent from the start.
The equation becomes $ax^2 + c = 0$.
Method — extract the root with ± discipline:
- Isolate $x^2$: $\;x^2 = -\dfrac{c}{a}$
- If $-c/a \ge 0$, take the square root: $x = \pm\sqrt{-c/a}$
- If $-c/a < 0$, no real roots.
Worked example: Solve $3x^2 - 48 = 0$.
$x^2 = \dfrac{48}{3} = 16 \;\Longrightarrow\; x = \pm 4.$
The equation becomes $ax^2 + bx = 0$.
Method — factor out the variable:
- $x(ax + b) = 0$
- Zero-product property: $x = 0$ or $ax + b = 0$
- So $x = 0$ or $x = -\dfrac{b}{a}$
Worked example: Solve $5x^2 - 20x = 0$.
$x(5x - 20) = 0 \;\Longrightarrow\; x = 0$ or $x = 4.$