Standard Form (Quadratic)
AlgebraStandard 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 is on one side, with the squared term first, the x-term second, and the number last, all equal to zero.
Example
$x^2 - 3x + 2 = 0$ is in standard form. $a = 1$, $b = -3$, $c = 2$. Compare: $2x^2 + 5x = 3$ is NOT yet standard - rearrange to $2x^2 + 5x - 3 = 0$.
Key Insight
Standard form is important because the quadratic formula needs $a$, $b$, and $c$ identified correctly. Always rearrange to standard form before applying the formula.
Definition
In standard form $ax^2 + bx + c = 0$, $a$ is the coefficient of $x^2$ (must be non-zero), $b$ is the coefficient of $x$ (can be zero), and $c$ is the constant term (can be zero). The form allows direct reading of coefficients for the quadratic formula and discriminant calculation.
Example
Convert $y = 3(x-1)^2 - 7$ to standard form: expand to $y = 3(x^2 - 2x + 1) - 7 = 3x^2 - 6x + 3 - 7 = 3x^2 - 6x - 4$. Set to zero: $3x^2 - 6x - 4 = 0$.
Key Insight
Standard form and vertex form ($a(x-h)^2 + k$) of a quadratic each have advantages. Standard form is best for the formula and discriminant; vertex form directly reveals the vertex and axis of symmetry.
Definition
The standard form $ax^2 + bx + c = 0$ can be reduced to the depressed quadratic $t^2 + pt + q = 0$ by the substitution $x = t - b/(2a)$, eliminating the linear term. The solutions of the depressed form are $t = (-p \pm \sqrt{p^2 - 4q})/2$. This reduction process generalizes to the Tschirnhaus transformation used to eliminate sub-leading terms in higher-degree polynomials.
Example
$2x^2 + 8x + 5 = 0$: substitute $x = t - 2$ to get $2(t-2)^2 + 8(t-2) + 5 = 0$, simplifying to $2t^2 - 3 = 0$, $t = \pm\sqrt{3/2}$. Then $x = t - 2$.
Key Insight
The reduction to depressed form is the historical technique that led to formulas for degree-$3$ and degree-$4$ polynomials (Cardano's and Ferrari's formulas). For degree $5$ and above, no such general formula exists (Abel-Ruffini theorem).