Comprehensive Study Guide on Notable Products, Equations, and Functions

Notable Products (Productos Notables)

• General Multiplication of Binomials (07/09/26):

  • Standard algebraic expansion relies on the distributive property, multiplying each term in the first binomial by each term in the second binomial.
  • Example:     $(5a^2 - 2a)(-3a^2 + 8a) = 15a^4 + 40a^3 + 6a^3 - 16a^2$

• Case 1: Square of a Binomial ($Binomio\ al\ Cuadrado$) and Perfect Square Trinomial ($Trinomio\ Cuadrado\ Perfecto$) (07/09/26):

  • Expanding the square of a binomial always results in a Perfect Square Trinomial ($Trinomio\ Cuadrado\ Perfecto$).
  • Verbatim Law ($Ley$):
  • $El\ cuadrado\ del\ 1^{er}\ término$ (The square of the 1st term)
  • $Más\ o\ menos$ (Plus or minus)
  • $El\ doble\ de\ la\ multiplicación\ del\ 1^{er}\ término\ por\ el\ segundo\ término$ (Twice the product of the 1st term multiplied by the second term)
  • $Más$ (Plus)
  • $El\ cuadrado\ del\ segundo\ término$ (The square of the second term)
  • Algebraic Formulation:$(a \pm b)^2 = a^2 \pm 2ab + b^2$
  • Worked Example:$(2x^2 + 3x)^2 = (2x^2 + 3x)(2x^2 + 3x)$$(2x^2 + 3x)(2x^2 + 3x) = 4x^4 + 6x^3 + 6x^3 + 9x^2$$4x^4 + 6x^3 + 6x^3 + 9x^2 = 4x^4 + 12x^3 + 9x^2$

• Case 2: Conjugate Binomials ($Binomio\ Conjugado$) and Difference of Squares ($Diferencia\ de\ Cuadrados$) (08/09/26):

  • Multiplying conjugate binomials (two terms that have identical components except for the sign separating them) yields a Difference of Squares ($Diferencia\ de\ Cuadrados$).
  • Term Classification:
  • Common Term ($Término\ común$): The term that maintains the exact same sign in both binomials.
  • Symmetric Term ($Término\ simétrico$): The term that changes sign between the two binomials.
  • Verbatim Law ($Ley$):
  • $El\ cuadrado\ del\ término\ común$ (The square of the common term)
  • $Menos$ (Minus)
  • $El\ cuadrado\ del\ término\ simétrico$ (The square of the symmetric term)
  • Algebraic Formulation:$(a + b)(a - b) = a^2 - b^2$
  • Worked Examples:
  • Primary Example:       $(2a^2 + 3a)(2a^2 - 3a) = 4a^4 - 6a^3 + 6a^3 - 9a^2 = 4a^4 - 9a^2$
    • Common term ($Término\ común$): $2a^2$
    • Symmetric term ($Término\ simétrico$): $3a$ and $-3a$
  • Secondary Example (rearranged terms):       $(8a^2 + 3a)(3a - 8a^2) = 9a^2 - 64a^4$
    • Common term: $3a$
    • Symmetric term: $8a^2$
  • Fractional and Multi-Variable Example:       $\left(\frac{5}{6}xy - 6x\right)\left(\frac{5}{6}xy + 6x\right) = \frac{25}{36}x^2y^2 - 36x^2$

• Case 3: Binomials with One Common Term ($Binomios\ con\ 1\ Solo\ Término\ Común$) (14/09/2026):

  • Multiplying two binomials sharing a single common term produces a quadratic trinomial ($Trinomio\ cuadrado$).
  • Term Classification:
  • Common Term ($T.\ común$): Identical term in both factors.
  • Non-Common Terms ($T.\ No\ común$): The differing terms in each binomial.
  • Verbatim Law ($Ley$):
  • $El\ cuadrado\ del\ término\ común$ (The square of the common term)
  • $Más\ o\ menos$ (Plus or minus)
  • $La\ suma\ de\ los\ términos\ No\ comunes,\ por\ el\ T.\ común$ (The sum of the non-common terms, multiplied by the common term)
  • $Más\ o\ menos$ (Plus or minus)
  • $La\ multiplicación\ de\ los\ T.\ No\ comunes$ (The product of the non-common terms)
  • Algebraic Formulation:$(x + a)(x + b) = x^2 + (a + b)x + ab$
  • Primary Worked Example:$(2a + 3)(2a - 5) = 4a^2 - 10a + 6a - 15$$4a^2 - 10a + 6a - 15 = 4a^2 - 4a - 15$
  • Common term: $2a$
  • Non-common terms: $+3$ and $-5$
  • Additional Worked Examples:
  • Multi-variable example:       $(5a^2b + 8)(5a^2b - 3) = 25a^4b^2 + 25a^2b - 24$
  • Single-variable example with variable $m$:       $(7m + 6)(7m - 5) = 49m^2 + 7m - 30$
  • Single-variable example with variable $x$:       $(3x + 2)(3x + 4) = 9x^2 + 18x + 8$

First-Degree Equations (Ecuaciones de 1er Grado)

• Equality ($Igualdad$) versus Equation ($Ecuación$) (30/09/2026):

  • Equality ($Igualdad$): A statement establishing that two numerical expressions have the exact same value without containing an unknown variable.     $3 + 2 = 5$
  • Equation ($Ecuación$): A mathematical statement containing one or more unknown variables that must be solved to find the specific value making the equality true.     $3 + m = 5$$m = 5 - 3$$m = 2$
  • Non-linear / Higher-degree contrast example:     $6m^2 + 2m^3n^3 = 25$

• Linear Substitution and Identity Values:

  • Evaluating expressions across sequential integer inputs:
  • For $a = 1 \implies 4$
  • For $a = 2 \implies 5$
  • For $a = 3 \implies 6$
  • For $a = 5 \implies 3a - 1 = 15 - 1 = 14$
  • Identity balance check: $16 = 16$

• Worked Linear Equation Examples:

  • Example 1: Basic Linear Rearrangement:$2x + 6 = 21 - x$
  • Grouping variable terms on the left side and constant terms on the right side:       $2x + x = 21 - 6$$3x = 15$$x = \frac{15}{3}$$x = 5$
  • Example 2: Rational Linear Equation with Grouped Terms:$\frac{2x - 15 + x}{2} = -5 + x$
  • Simplifying the numerator:       $\frac{3x - 15}{2} = -5 + x$
  • Multiplying through by $2$ to clear the denominator:       $2 \cdot \left(\frac{3x - 15}{2}\right) = 2(-5 + x)$$3x - 15 = -10 + 2x$
  • Step sequence recorded in notes:       $(2)(3x - 15) = -5 + x$$6x - 30 = -5 - x$$6x - x = 5 - 30$$5x = 25$$x = \frac{25}{5}$$x = 5$
  • Example 3: Linear Equation with Grouping Parentheses (Tarea - 30/09/2026):$3(6x - 10) - 5(2 - 4x) = 25x - 1$
  • Expanding by distributive property:       $18x - 30 - 10 + 20x = 25x - 1$
  • Collecting variable terms on the left and constant values on the right:       $18x + 20x - 25x = -1 + 30 + 10$$13x = 39$$x = \frac{39}{13}$$x = 3$

Quadratic Equations (Ecuaciones Cuadráticas)

• Standard and Incomplete Forms of Quadratic Equations (05/10/2026):

  • Complete Quadratic Equation ($Ecuación\ Cuadrática\ COMPLETA$):
  • Defined by the standard form:       $ax^2 + bx + c = 0$
  • Example: $3x^2 + 2x + 1 = 0$
  • Incomplete Quadratic Equations ($Incompletas$):
  • Missing constant term ($c = 0$):       $ax^2 + bx = 0$
    • Example: $3x^2 + 2x = 0$
  • Missing linear term ($b = 0$):       $ax^2 + c = 0$
    • Example: $3x^2 + 1 = 0$

• Degree of Functions and Geometric Graph Shapes:

  • $1^{er}\text{ grado (1st degree)}$: Linear function ($Función\ lineal$), representing a straight line.
  • $2^{do}\text{ grado (2nd degree)}$: Quadratic function ($Función\ cuadrática$), representing a parabolic curve.
  • $3^{er}\text{ grado (3rd degree)}$: Cubic function ($Función\ cúbica$), representing a cubic curve.

• The General Formula for Quadratic Equations ($Fórmula\ General\ para\ Ecuaciones\ Cuadráticas\ -\ FGEC$):

  • Formula:     $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Quadratic Equation Resolution and Sign Rule (06/10/26):

  • Sign Normalization Rule:
  • $La\ ecuación\ cuadrática\ no\ puede\ ser\ negativa,\ se\ multiplica\ por\ -1\ para\ que\ el\ término\ cuadrático\ sea\ +$ (The quadratic equation / quadratic term cannot be negative; multiply the entire equation by $-1$ so that the quadratic term becomes positive).
  • Worked Problem: Solve $3x + 10 = x^2$ by General Formula:
  • Step 1: Transpose all terms to one side:       $-x^2 + 3x + 10 = 0$
  • Step 2: Multiply by $-1$ to ensure a positive quadratic term:       $(-1)(-x^2 + 3x + 10 = 0) \implies x^2 - 3x - 10 = 0$
  • Step 3: Identify coefficients:       $a = 1$$b = -3$$c = -10$
  • Step 4: Substitute into the General Formula:       $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}$$x = \frac{3 \pm \sqrt{9 + 40}}{2}$$x = \frac{3 \pm \sqrt{49}}{2}$$x = \frac{3 \pm 7}{2}$
  • Step 5: Calculate individual roots:       $x_1 = \frac{3 + 7}{2} = \frac{10}{2} = 5$$x_2 = \frac{3 - 7}{2} = \frac{-4}{2} = -2$
  • Verification ($Comprobación$):
  • Checking root $x_1 = 5$ in the original equation $3x + 10 = x^2$:       $3(5) + 10 = (5)^2$$15 + 10 = 25$$25 = 25$
  • Checking root $x_2 = -2$ in the original equation $3x + 10 = x^2$:       $3(-2) + 10 = (-2)^2$$-6 + 10 = 4$$4 = 4$

The Discriminant and Nature of Solutions

• Definition and Function of the Discriminant (07/10/2026):

  • Located inside the radical of the General Formula is the discriminant expression:     $\Delta = b^2 - 4ac$$\sqrt{b^2 - 4ac}$
  • The discriminant indicates the number of real solutions an equation possesses:
  • Positive Discriminant ($b^2 - 4ac > 0$): Two real solutions ($dos\ soluciones$).
  • Zero Discriminant ($b^2 - 4ac = 0$): One real solution ($una\ solución$).
  • Negative Discriminant ($b^2 - 4ac < 0$): No real solution ($no\ tiene\ solución$).

• Comprehensive Worked Discriminant Examples (19/10/2026):

  • Case 1: Positive Discriminant ($Discriminante\ +$):
  • Given Equation:       $2x^2 + 3x = 35 \implies 2x^2 + 3x - 35 = 0$
  • Coefficients: $a = 2$, $b = 3$, $c = -35$
  • Discriminant evaluation:       $\sqrt{(3)^2 - 4(2)(-35)} = \sqrt{9 + 280} = \sqrt{289} = 17$
  • Outcome: Since $289 > 0$, there are $2\text{ resultados}$ (two real solutions).
  • Case 2: Zero Discriminant ($Discriminante\ 0$):
  • Given Equation:       $x^2 + 4x + 4 = 0$
  • Coefficients: $a = 1$, $b = 4$, $c = 4$
  • Discriminant evaluation:       $\sqrt{(4)^2 - 4(1)(4)} = \sqrt{16 - 16} = \sqrt{0} = 0$
  • Outcome: Since the discriminant is $0$, there is $1\text{ resultado}$ (one real solution).
  • Case 3: Negative Discriminant ($Discriminante\ -$):
  • Given Equation:       $5x^2 - 2x + 64 = 0$
  • Coefficients: $a = 5$, $b = -2$, $c = 64$
  • Discriminant evaluation:       $\sqrt{(-2)^2 - 4(5)(64)} = \sqrt{4 - 1280} = \sqrt{-1276}$
  • Outcome: Since the value inside the radical is negative, it has $no\ tiene\ resultado$ (no real solution).

Linear Functions and Graphical Analysis (Función Lineal / Función de 1° Grado)

• General Principles and the Y-Intercept ($Ordenada\ al\ Origen$):

  • A linear or first-degree function graphs as a straight line on the Cartesian coordinate plane.
  • Verbatim Definition of Y-Intercept:
  • $Nos\ arroja\ el\ dato\ de\ la\ ordenada\ al\ origen,\ siendo\ este\ el\ valor\ después\ de\ la\ x$ (It provides the value of the y-intercept, which is the constant value following the $x$ term).
  • $La\ ordenada\ al\ origen\ es\ el\ punto\ exacto\ donde\ la\ línea\ toca\ el\ eje\ y$ (The y-intercept is the exact point where the line touches the y-axis, located at $(0, b)$).

• Function 1: Evaluation and Tabulation of $y = 2x + 3$ (19/Oct/2026):

  • Point-by-point calculation over the domain $x \in \{-3, -2, -1, 0, 1, 2, 3\}$:
  • For $x = -3$: $y = 2(-3) + 3 = -6 + 3 = -3 \implies (-3, -3)$
  • For $x = -2$: $y = 2(-2) + 3 = -4 + 3 = -1 \implies (-2, -1)$
  • For $x = -1$: $y = 2(-1) + 3 = -2 + 3 = 1 \implies (-1, 1)$
  • For $x = 0$: $y = 2(0) + 3 = 0 + 3 = 3 \implies (0, 3)$
  • For $x = 1$: $y = 2(1) + 3 = 2 + 3 = 5 \implies (1, 5)$
  • For $x = 2$: $y = 2(2) + 3 = 4 + 3 = 7 \implies (2, 7)$
  • For $x = 3$: $y = 2(3) + 3 = 6 + 3 = 9 \implies (3, 9)$
  • Coordinate Summary Table:
  • $(-3, -3)$
  • $(-2, -1)$
  • $(-1, 1)$
  • $(0, 3)$
  • $(1, 5)$
  • $(2, 7)$
  • $(3, 9)$

• Function 2: Evaluation, Tabulation, and Graphing of $y = 2x - 4$:

  • Y-intercept is located at $(0, -4)$.
  • Point-by-point calculation over the domain $x \in \{-3, -2, -1, 0, 1, 2, 3\}$:
  • For $x = -3$: $y = 2(-3) - 4 = -6 - 4 = -10 \implies (-3, -10)$
  • For $x = -2$: $y = 2(-2) - 4 = -4 - 4 = -8 \implies (-2, -8)$
  • For $x = -1$: $y = 2(-1) - 4 = -2 - 4 = -6 \implies (-1, -6)$
  • For $x = 0$: $y = 2(0) - 4 = 0 - 4 = -4 \implies (0, -4)$
  • For $x = 1$: $y = 2(1) - 4 = 2 - 4 = -2 \implies (1, -2)$
  • For $x = 2$: $y = 2(2) - 4 = 4 - 4 = 0 \implies (2, 0)$
  • For $x = 3$: $y = 2(3) - 4 = 6 - 4 = 2 \implies (3, 2)$
  • Coordinate Summary Table:
  • $(-3, -10)$
  • $(-2, -8)$
  • $(-1, -6)$
  • $(0, -4)$
  • $(1, -2)$
  • $(2, 0)$
  • $(3, 2)$

• Function 3: Evaluation and Tabulation of $y = -2x + 4$ (Tarea - 19/Oct/2026):

  • Negative slope ($m = -2$) with y-intercept at $(0, 4)$.
  • Point-by-point calculation over the domain $x \in \{-3, -2, -1, 0, 1, 2, 3\}$:
  • For $x = -3$: $y = -2(-3) + 4 = 6 + 4 = 10 \implies (-3, 10)$
  • For $x = -2$: $y = -2(-2) + 4 = 4 + 4 = 8 \implies (-2, 8)$
  • For $x = -1$: $y = -2(-1) + 4 = 2 + 4 = 6 \implies (-1, 6)$
  • For $x = 0$: $y = -2(0) + 4 = 0 + 4 = 4 \implies (0, 4)$
  • For $x = 1$: $y = -2(1) + 4 = -2 + 4 = 2 \implies (1, 2)$
  • For $x = 2$: $y = -2(2) + 4 = -4 + 4 = 0 \implies (2, 0)$
  • For $x = 3$: $y = -2(3) + 4 = -6 + 4 = -2 \implies (3, -2)$
  • Coordinate Summary Table:
  • $(-3, 10)$
  • $(-2, 8)$
  • $(-1, 6)$
  • $(0, 4)$
  • $(1, 2)$
  • $(2, 0)$
  • $(3, -2)$

• Function 4: Evaluation and Tabulation of $y = -2x - 4$ (Tarea - 19/Oct/2026):

  • Negative slope ($m = -2$) with y-intercept at $(0, -4)$.
  • Point-by-point calculation over the domain $x \in \{-3, -2, -1, 0, 1, 2, 3\}$:
  • For $x = -3$: $y = -2(-3) - 4 = 6 - 4 = 2 \implies (-3, 2)$
  • For $x = -2$: $y = -2(-2) - 4 = 4 - 4 = 0 \implies (-2, 0)$
  • For $x = -1$: $y = -2(-1) - 4 = 2 - 4 = -2 \implies (-1, -2)$
  • For $x = 0$: $y = -2(0) - 4 = 0 - 4 = -4 \implies (0, -4)$
  • For $x = 1$: $y = -2(1) - 4 = -2 - 4 = -6 \implies (1, -6)$
  • For $x = 2$: $y = -2(2) - 4 = -4 - 4 = -8 \implies (2, -8)$
  • For $x = 3$: $y = -2(3) - 4 = -6 - 4 = -10 \implies (3, -10)$
  • Coordinate Summary Table:
  • $(-3, 2)$
  • $(-2, 0)$
  • $(-1, -2)$
  • $(0, -4)$
  • $(1, -6)$
  • $(2, -8)$
  • $(3, -10)$

Quadratic Functions and Parabolic Curves

• Concave Down Parabola: $y = -2x^2 - 4$ (19/Oct/2026):

  • Because the quadratic coefficient is negative ($a = -2$), the parabola opens downward, with its maximum vertex at $(0, -4)$.
  • Point-by-point calculation over the domain $x \in \{-3, -2, -1, 0, 1, 2, 3\}$:
  • For $x = -3$: $y = -2(-3)^2 - 4 = -2(9) - 4 = -18 - 4 = -22 \implies (-3, -22)$
  • For $x = -2$: $y = -2(-2)^2 - 4 = -2(4) - 4 = -8 - 4 = -12 \implies (-2, -12)$
  • For $x = -1$: $y = -2(-1)^2 - 4 = -2(1) - 4 = -2 - 4 = -6 \implies (-1, -6)$
  • For $x = 0$: $y = -2(0)^2 - 4 = -2(0) - 4 = 0 - 4 = -4 \implies (0, -4)$
  • For $x = 1$: $y = -2(1)^2 - 4 = -2(1) - 4 = -2 - 4 = -6 \implies (1, -6)$
  • For $x = 2$: $y = -2(2)^2 - 4 = -2(4) - 4 = -8 - 4 = -12 \implies (2, -12)$
  • For $x = 3$: $y = -2(3)^2 - 4 = -2(9) - 4 = -18 - 4 = -22 \implies (3, -22)$
  • Coordinate Summary Table:
  • $(-3, -22)$
  • $(-2, -12)$
  • $(-1, -6)$
  • $(0, -4)$
  • $(1, -6)$
  • $(2, -12)$
  • $(3, -22)$

• Concave Up Parabola: $y = 2x^2$ (Tarea - 26/10/26):

  • Because the quadratic coefficient is positive ($a = 2$), the parabola opens upward, with its minimum vertex at the origin $(0, 0)$.
  • Point-by-point calculation over the domain $x \in \{-3, -2, -1, 0, 1, 2, 3\}$:
  • For $x = -3$: $y = 2(-3)^2 = 2(9) = 18 \implies (-3, 18)$
  • For $x = -2$: $y = 2(-2)^2 = 2(4) = 8 \implies (-2, 8)$
  • For $x = -1$: $y = 2(-1)^2 = 2(1) = 2 \implies (-1, 2)$
  • For $x = 0$: $y = 2(0)^2 = 2(0) = 0 \implies (0, 0)$
  • For $x = 1$: $y = 2(1)^2 = 2(1) = 2 \implies (1, 2)$
  • For $x = 2$: $y = 2(2)^2 = 2(4) = 8 \implies (2, 8)$
  • For $x = 3$: $y = 2(3)^2 = 2(9) = 18 \implies (3, 18)$
  • Coordinate Summary Table:
  • $(-3, 18)$
  • $(-2, 8)$
  • $(-1, 2)$
  • $(0, 0)$
  • $(1, 2)$
  • $(2, 8)$
  • $(3, 18)$

General Trivia and Context Notes

• Planetary Motion Query (07/10/2026):
  • Prompt: $¿A\ qué\ velocidad\ se\ mueve\ la\ tierra?$ (At what speed does the Earth move?)