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Quadratic Roots
Standard Form and the Parabola
A quadratic equation is a second-degree polynomial in standard form $ax^2 + bx + c = 0$, representing a parabola.
$$ax^2 + bx + c = 0$$
The standard form where 'a' determines the curve's opening direction.
$a$=Leading coefficient (quadratic term)(constant)
$b$=Linear coefficient(constant)
$c$=Constant term (y-intercept)(constant)
$a > 0$
→Parabola opens upwards (U-shape)
$a < 0$
→Parabola opens downwards (inverted U)
Geometric Meaning: The roots are the x-values where the curve intersects the horizontal axis.
Roots vs Intercepts: In mathematics, the terms roots, zeros, and x-intercepts are often used interchangeably for these values.
Predicting with the Discriminant
The discriminant ($D$) determines the nature of the roots without solving the entire equation.
$$D = b^2 - 4ac$$
The part of the quadratic formula under the radical.
$D$=Discriminant(unitless)
$D > 0$
→Two real and distinct roots
$D = 0$
→One real and equal root (repeated)
$D < 0$
→Two complex / imaginary roots
Perfect Squares: If $a, b, c$ are rational and $D$ is a perfect square, the roots are rational.
Irrational Roots: If $D > 0$ but not a perfect square (with rational $a, b, c$), the roots are irrational and come in conjugate pairs like $p \pm \sqrt{q}$.
Complex Conjugates: When $D < 0$, complex roots always appear in conjugate pairs $p \pm qi$.
Negative Reality: A negative $D$ implies the parabola never touches the x-axis, existing entirely above or below it.
The Quadratic Formula
The Quadratic Formula provides the exact values for the roots using the coefficients $a, b,$ and $c$.
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Derives solutions for any quadratic equation.
$\pm$=Plus-minus sign(indicates two solutions)
$\sqrt{...}$=[radical] expression(operation)
$\sqrt{D} = 0$
→$x = -b / 2a$ (vertex position)
Sign Safety: Always keep the sign of $b$ consistent; $-b$ means 'opposite of $b$'.
Calculation Order: Always compute the discriminant $D$ first to verify if real roots exist before proceeding.
Vieta's Formulas
Vieta's formulas relate the sum and product of the roots directly to the coefficients.
$$\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}$$
Bypasses finding individual roots to find their collective properties.
$\alpha, \beta$=The two roots of the equation(x-values)
$\alpha + \beta$=Sum of roots(unitless)
$\alpha\beta$=Product of roots(unitless)
$a = 1$
→monic quadratic: Sum = -b, Product = c
Monic Advantage: In a monic equation ($x^2 + px + q = 0$), the sum is simply the negative coefficient of $x$.
Constructing Equations: Given roots $\alpha, \beta$, the equation is $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.
Symmetric Expressions: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ lets you compute higher expressions without finding individual roots.
Difference of Roots: $|\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} = \frac{\sqrt{D}}{|a|}$.
Symmetry: These formulas hold true even for complex roots.
Cube Roots of Unity
The cube roots of unity are the three solutions to $x^3 = 1$, consisting of one real root and two complex conjugate roots.
$$1, \quad \omega = \frac{-1 + \sqrt{3}i}{2}, \quad \omega^2 = \frac{-1 - \sqrt{3}i}{2}$$
Found by solving $x^3 - 1 = 0 \implies (x-1)(x^2+x+1) = 0$.
$\omega$=Complex cube root of unity (omega)
$\omega^2$=Second complex cube root
$n$ is a multiple of 3
→$\omega^n = 1$
$n \equiv 1 \pmod{3}$
→$\omega^n = \omega$
$n \equiv 2 \pmod{3}$
→$\omega^n = \omega^2$
Sum Property: $1 + \omega + \omega^2 = 0$, equivalently $\omega + \omega^2 = -1$.
Product Property: $\omega^3 = 1$ (product of all three roots is unity).
Conjugate Pair: $\omega$ and $\omega^2$ are complex conjugates of each other.
Exponent Reduction: For any integer $n$, divide exponent by 3 and use the remainder: $\omega^{11} = \omega^{9} \cdot \omega^{2} = (\omega^3)^3 \cdot \omega^2 = \omega^2$.
Factorization: $x^3 + y^3 = (x+y)(x+\omega y)(x+\omega^2 y)$ and $x^3 - y^3 = (x-y)(x-\omega y)(x-\omega^2 y)$.
Fourth Roots of Unity
The fourth roots of unity are the four solutions to $x^4 = 1$, consisting of two real and two imaginary values.
$$+1, \quad -1, \quad +i, \quad -i$$
Found by solving $x^4 - 1 = 0 \implies (x^2-1)(x^2+1) = 0$.
$\pm 1$=Real fourth roots
$\pm i$=Imaginary fourth roots
Sum Property: $1 + (-1) + i + (-i) = 0$ (sum of all fourth roots is zero).
Product Property: $1 \times (-1) \times i \times (-i) = -1$ (product of all fourth roots is $-1$, unlike cube roots).
Real Roots: $+1$ and $-1$ are additive inverses.
Imaginary Roots: $+i$ and $-i$ are complex conjugates.
Finding $n$th Roots: For $\sqrt[4]{a}$, multiply each fourth root of unity by $\sqrt[4]{|a|}$. For example, $\sqrt[4]{16} = 2 \times \{1, -1, i, -i\} = \{2, -2, 2i, -2i\}$.