Mathematical Facts, Constants & Formulas
Grade 9: Polynomial Factoring, Quadratic Theory & Exponential Functions
~9 min read
HESTEN ACADEMIC REFERENCE ARCHIVE
CALL NO: QA152.H47 2026 • DEWEY: 512.9 • VOL. IX
Volume IX • Classical Algebra I Compendium
Ninth Grade Mathematics Reference Codex (Algebra I)
The Complete Scholastic Guide to Polynomial Arithmetic, Master $ac$-Factoring, Quadratic Theory & Discriminant ($\Delta$), Parabolic Geometry, Exponential Models, Sequences & Synoptic Concordance
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An exhaustive reference codex for secondary algebra scholars, instructors, and pre-collegiate mathematicians. Systematizing polynomial ring operations, prime quadratic factorization algorithms, the general quadratic formula, parabolic axis and extrema theorems, discrete arithmetic/geometric progressions, and computational mental algebra heuristics.
§ 9.1
Polynomial Arithmetic & Binomial FOIL Expansion
FOIL Expansion Law
$$(a + b)(c + d) = \underbrace{ac}_{\text{First}} + \underbrace{ad}_{\text{Outer}} + \underbrace{bc}_{\text{Inner}} + \underbrace{bd}_{\text{Last}}$$
$(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$.
Degree & Polynomial Type
$$\text{deg}(P(x)) = \text{Highest Exponent Power of Variable}$$
Deg 0: Constant • Deg 1: Linear • Deg 2: Quadratic • Deg 3: Cubic • Deg 4: Quartic.
§ 9.2
Factoring Quadratic Trinomials: The Master $ac$-Method
ALGORITHM 9.2.1
Topic: The Complete $ac$-Method Algorithm
To factor general trinomial $ax^2 + bx + c$:
- Factor out any common greatest common factor ($\text{GCF}$).
- Compute product $P = a \cdot c$ and target sum $S = b$.
- Find two numbers $m, n$ such that $m \cdot n = a \cdot c$ and $m + n = b$.
- Rewrite middle term: $ax^2 + mx + nx + c$, then factor by grouping!
§ 9.3
Special Factoring Identities & Conjugate Pairs
Difference of Two Squares
$$a^2 - b^2 = (a + b)(a - b)$$
$9x^2 - 49 = (3x + 7)(3x - 7)$. (Sum of squares $a^2+b^2$ is prime over reals).
Perfect Square Trinomial (+)
$$a^2 + 2ab + b^2 = (a + b)^2$$
$x^2 + 12x + 36 = (x + 6)^2$.
Perfect Square Trinomial (-)
$$a^2 - 2ab + b^2 = (a - b)^2$$
$4x^2 - 20x + 25 = (2x - 5)^2$.
§ 9.4
The Quadratic Formula & Discriminant ($\Delta$) Analysis
THEOREM 9.4.1
Topic: The General Quadratic Resolution
For any quadratic equation $ax^2 + bx + c = 0$ with $a \neq 0$:
$$x = \frac{-b \pm \sqrt{\Delta}}{2a} \quad \text{where Discriminant } \Delta = b^2 - 4ac$$
| Discriminant Value ($\Delta$) | Nature of Roots | Parabola Real $x$-Intercepts |
|---|---|---|
| $\Delta > 0$ (Perfect Square) | Two distinct rational roots | Crosses $x$-axis twice at rational integers/fractions |
| $\Delta > 0$ (Non-Square) | Two distinct irrational conjugate roots | Crosses $x$-axis twice at real radical values |
| $\Delta = 0$ (Zero) | Exactly one repeated real root (double root) | Parabola vertex touches $x$-axis tangentially |
| $\Delta < 0$ (Negative) | Two complex conjugate roots ($a \pm bi$) | Parabola floats entirely above or below $x$-axis |
EXEMPLUM 9.4
Quadratic Equation Resolution
Problem: Solve $2x^2 - 5x - 3 = 0$ using the Quadratic Formula.
Step 1 (Identify Coefficients): $a = 2, b = -5, c = -3$.
Step 2 (Compute Discriminant): $\Delta = (-5)^2 - 4(2)(-3) = 25 - (-24) = 49 = 7^2$ (Two rational roots!).
Step 3 (Evaluate Roots): $x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4}$.
Step 4 (Separate Branches): $x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3 \quad \vert \quad x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$.
Result: $x = \left\{3, -\frac{1}{2}\right\}$.
■ Q.E.D.
§ 9.5
Parabolic Geometry: Vertex Form & Extrema
Form $y = a(x - h)^2 + k$
$$\text{Vertex } V = (h, k)$$
Opens upwards (minimum) if $a > 0$; opens downwards (maximum) if $a < 0$.
Vertical Axis of Symmetry
$$x = -\frac{b}{2a}$$
Vertical line passing directly through the vertex coordinate $h$.
§ 9.6
Exponential Growth, Decay & Compound Interest
Exponential Growth ($r > 0$)
$$y = a(1 + r)^t$$
Population expansions, viral propagation, price inflation.
Exponential Decay ($r > 0$)
$$y = a(1 - r)^t$$
Radioactive half-life ($y = a(0.5)^{t/h}$), asset depreciation.
Compound Interest
$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
$P = \text{Principal}, r = \text{Annual rate}, n = \text{Compounds per yr}, t = \text{Years}$.
§ 9.7
Arithmetic & Geometric Sequences ($a_n$)
Arithmetic Sequence ($n^{\text{th}}$ Term)
$$a_n = a_1 + (n - 1)d$$
$d = \text{common difference } (a_{k+1} - a_k)$. Constant linear slope.
Geometric Sequence ($n^{\text{th}}$ Term)
$$a_n = a_1 \cdot r^{n-1}$$
$r = \text{common ratio } \left(\frac{a_{k+1}}{a_k}\right)$. Exponential growth factor.
§ 9.8
Linear Systems by Elimination & 2D Inequalities
Linear Elimination Protocol
Multiply one or both equations by non-zero constants so that coefficients of one variable are exact additive opposites ($+k$ and $-k$). Add equations together to eliminate the variable!
- Graphing 2D Inequalities: Solid boundary line for $\le, \ge$; Dashed line for $<, >$.
- Test Point: Plug in $(0, 0)$ to decide which half-plane to shade.
§ 9.9
Statistics: Standard Deviation ($\sigma$) & Correlation ($r$)
| Metric | Range | Statistical Meaning |
|---|---|---|
| Correlation ($r$) | $-1 \le r \le +1$ | Strength and direction of linear association ($+1$ perfect positive, $-1$ perfect negative, $0$ none) |
| Standard Deviation ($\sigma$) | $\sigma \ge 0$ | Measure of typical data spread dispersion around the mean $\bar{x}$ |
| Residual | $\mathbb{R}$ | $\text{Residual} = \text{Observed } y - \text{Predicted } \hat{y}$ (Random scatter indicates good linear fit) |
§ 9.10
Scholia & Computational Mental Math Speed Hacks
HACK I
Vieta's Formula Root Shortcuts
$$r_1 + r_2 = -\frac{b}{a} \quad \vert \quad r_1 \cdot r_2 = \frac{c}{a}$$
Never solve a quadratic just to find the sum or product of its roots on an exam. Read them off coefficients in 1 second!
HACK II
The Diamond X-Factoring Hack
$$\text{Top: } a \cdot c \quad \vert \quad \text{Bottom: } b \implies \text{Wings: } m, n$$
Draw a big X. Top is product $ac$, bottom is sum $b$. Filling left and right gives your binomial factors instantly.
HACK III
Quick Discriminant Mental Check
$$\text{If } a \text{ and } c \text{ have opposite signs} \implies \Delta > 0 \text{ ALWAYS!}$$
Because $-4ac$ will be positive, $b^2 + \text{positive} > 0$. Guaranteed 2 real roots without computing!
HACK IV
Parabola Step Pattern (1a, 3a, 5a)
$$\text{From vertex: Over } 1 \to \text{Up } 1a, \ \text{Over } 1 \to \text{Up } 3a, \ \text{Over } 1 \to \text{Up } 5a$$
Graph any parabola instantaneously without making an input-output table!
§ 9.11
The Grand Synoptic Tables & Complete Grade 9 / Algebra I Student Cheat Sheet
Grade 9 / Algebra I Master Reference Concordance
Authorized curriculum reference concordance • Algebra I Standards Complete
Quadratic Formula
- $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
- $\Delta = b^2 - 4ac$
- $\Delta > 0$: 2 Real roots
- $\Delta = 0$: 1 Repeated root
- $\Delta < 0$: 2 Complex roots
- Vieta: $r_1+r_2 = -b/a, r_1 r_2 = c/a$
Factoring Identities
- $a^2 - b^2 = (a+b)(a-b)$
- $(a+b)^2 = a^2 + 2ab + b^2$
- $(a-b)^2 = a^2 - 2ab + b^2$
- $ac$-Method: Split middle term
- Always factor out GCF first!
Parabola Features
- Axis of Symmetry: $x = -\frac{b}{2a}$
- Vertex: $(h, k)$ in $y=a(x-h)^2+k$
- $a > 0$: Opens up (Minimum)
- $a < 0$: Opens down (Maximum)
- Step pattern: $1a, 3a, 5a$
Exponentials
- Growth: $y = a(1+r)^t$
- Decay: $y = a(1-r)^t$
- Compound: $A = P(1 + \frac{r}{n})^{nt}$
- Half-life: $y = a(0.5)^{t/h}$
- Horizontal asymptote: $y = 0$
Sequences
- Arithmetic: $a_n = a_1 + (n-1)d$
- Arithmetic Difference: $d = a_k - a_{k-1}$
- Geometric: $a_n = a_1 \cdot r^{n-1}$
- Geometric Ratio: $r = a_{k} / a_{k-1}$
Statistics
- Correlation $r$: $[-1, +1]$
- $r \approx \pm 1$: Strong linear
- Residual: $y - \hat{y}$
- Standard Deviation: $\sigma$
- Linear regression line: $\hat{y} = mx + b$