Standard Form (Quadratic)
Standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a$ is not zero.
Formula
ax^2 + bx + c = 0
Definition
Standard form of a quadratic is $ax^2 + bx + c = 0$: everything on one side, the squared term first, the x-term second, the constant last, all equal to zero, with $a$ non-zero, $b$ possibly zero, and $c$ possibly zero. This form lets you read off coefficients directly for the quadratic formula and discriminant, while vertex form $a(x-h)^2 + k$ instead directly reveals the vertex and axis of symmetry. The standard form can be reduced to the depressed quadratic $t^2 + pt + q = 0$ by the substitution $x = t - b/(2a)$, eliminating the linear term, with solutions $t = (-p \pm \sqrt{p^2 - 4q})/2$; this reduction generalizes to the Tschirnhaus transformation used to eliminate sub-leading terms in higher-degree polynomials.
Example
$x^2 - 3x + 2 = 0$ is already in standard form, with $a = 1$, $b = -3$, $c = 2$, while $2x^2 + 5x = 3$ must be rearranged to $2x^2 + 5x - 3 = 0$ first. Converting $y = 3(x-1)^2 - 7$ to standard form: expand to $3x^2 - 6x + 3 - 7 = 3x^2 - 6x - 4$, so the equation is $3x^2 - 6x - 4 = 0$. For $2x^2 + 8x + 5 = 0$, substituting $x = t - 2$ gives $2t^2 - 3 = 0$, so $t = \pm\sqrt{3/2}$.
Key Insight
Standard form matters because the quadratic formula needs $a$, $b$, and $c$ identified correctly, so always rearrange into this form first. The reduction to depressed form is the historical technique that led to solution formulas for degree-$3$ and degree-$4$ polynomials (Cardano's and Ferrari's formulas); for degree $5$ and above, no such general formula exists at all, a fact known as the Abel-Ruffini theorem.