Chapter Review
Algebra
Algebraic Expressions and Simplification · Linear Equations · Quadratic Equations · Linear Inequalities · Word Problems and Problem Solving · Basic Matrices
Algebraic Expressions and Identities
Expressions combine variables, constants, and operations. Core identities — binomial squares, difference of squares, sum/difference of cubes — are the shortcuts for expanding and factoring.
Key Points
- •Like terms share the same variable part; combine by adding coefficients
- •$(a+b)^2 = a^2 + 2ab + b^2$ — never skip the middle term $2ab$
- •$a^2 - b^2 = (a+b)(a-b)$ — applies whenever two perfect squares are subtracted
- •$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$ — use SOAP: Same sign, Opposite, Always Positive
- •Always factor out the GCF before attempting any other factorization method
- •Algebraic fractions: factor numerator and denominator first, then cancel only complete factors
Formula
$$(a+b)^2 = a^2 + 2ab + b^2, \quad a^2 - b^2 = (a+b)(a-b)$$
Factorization Techniques
Factorization reverses expansion to reveal an expression's structure. Methods include GCF extraction, grouping (for 4+ terms), the AC method (splitting the middle term), and recognizing identity patterns.
Key Points
- •GCF first — always extract the greatest common factor before other methods
- •AC method: for $ax^2 + bx + c$, find two numbers with product $ac$ and sum $b$, then split the middle term
- •Grouping pairs terms to reveal a shared binomial factor: $x^2(x+3) - 4(x+3) = (x^2-4)(x+3)$
- •Perfect square trinomial check: middle term $= 2\sqrt{\text{first} \times \text{last}}$
- •After grouping, always check if the remaining quadratic factors further
Linear Equations and Systems
Linear equations ($ax + b = 0$) have one solution; two-variable systems ($ax + by = c$) are solved by substitution or elimination. Slope-intercept form $y = mx + c$ reveals the line's steepness and intercept.
Key Points
- •Transpose rule: moving a term across the equals sign flips its sign
- •Clear fractions by multiplying every term by the LCD before solving
- •Substitution: isolate one variable, plug into the other equation — best when one coefficient is 1
- •Elimination: multiply to match coefficients, then add/subtract — best when both equations are in standard form
- •Parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ → no solution
- •Coincident lines: all ratios equal → infinitely many solutions
Formula
$$y = mx + c, \quad m = -\frac{a}{b} \text{ from } ax + by + c = 0$$
Quadratic Equations — Solving Methods
Quadratics ($ax^2 + bx + c = 0$, $a \neq 0$) are solved by factorization (fastest when roots are rational) or the quadratic formula (always works). The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots without solving.
Key Points
- •Factorization: split the middle term, factor by grouping, apply zero-product property
- •Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ — universal solver
- •$\Delta > 0$: two distinct real roots; $\Delta = 0$: one repeated root; $\Delta < 0$: no real roots
- •If $\Delta$ is a perfect square, roots are rational — factorization should work
- •In word problems, always reject the root that violates physical constraints (negative length, negative time)
- •Divide the entire equation by the GCD of coefficients before applying the formula
Formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Vieta's Formulas and Forming Equations
Vieta's formulas link the sum and product of roots directly to the coefficients, enabling equation construction and answer verification without solving for individual roots.
Key Points
- •Sum of roots: $\alpha + \beta = -\frac{b}{a}$ — the negative sign is the most common error
- •Product of roots: $\alpha\beta = \frac{c}{a}$ — sign reveals whether roots share the same sign
- •If $\alpha\beta > 0$, roots have the same sign; if $\alpha\beta < 0$, roots have opposite signs
- •To form an equation from roots: $x^2 - (\alpha+\beta)x + \alpha\beta = 0$
- •For modified roots (e.g., $\alpha+k$ and $\beta+k$), compute the new sum and product from scratch
Formula
$$\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}$$
Linear Inequalities and Optimization
Inequalities behave like equations except when multiplying or dividing by a negative number, which reverses the sign. Linear programming finds the optimum of a linear function over a feasible region defined by inequality constraints.
Key Points
- •Multiplying/dividing by a negative flips the inequality sign: $a < b$ and $c < 0$ implies $ac > bc$
- •Strict inequalities ($<, >$) use dashed boundary lines; non-strict ($\le, \ge$) use solid lines
- •Test point method: substitute $(0,0)$ to determine which half-plane to shade
- •Corner Point Theorem: the optimum of a linear objective over a convex region occurs at a vertex
- •Feasible region = intersection of all constraint half-planes, restricted to $x \ge 0, y \ge 0$
- •Evaluate the objective function at every corner point — the largest is max, smallest is min
Formula
$$a < b \text{ and } c < 0 \implies ac > bc$$
Word Problem Models
Standard word problem archetypes — distance-speed-time, work-rate, mixture/alligation, age, and partnership — each have a core equation and a systematic setup procedure.
Key Points
- •D = S × T: convert all units consistently before setting up the equation
- •Average speed over equal distances: $S_{\text{avg}} = \frac{2S_1 S_2}{S_1 + S_2}$ (harmonic mean, not arithmetic)
- •Combined work: $\frac{1}{T_1} + \frac{1}{T_2} = \frac{1}{T_{\text{together}}}$, or shortcut $T = \frac{T_1 \cdot T_2}{T_1 + T_2}$
- •Alligation cross: ratio $= (C_{\text{mix}} - C_2) : (C_1 - C_{\text{mix}})$ — mixture value must lie between components
- •Partnership: profit ratio = (investment × time) ratio for each partner
- •Clock angle: $\theta = |30H - 5.5M|$; train crossing: $T = \frac{L_{\text{train}} + L_{\text{object}}}{S_{\text{relative}}}$
Formula
$$T_{\text{together}} = \frac{T_1 \cdot T_2}{T_1 + T_2}, \quad \frac{Q_1}{Q_2} = \frac{C_{\text{mix}} - C_2}{C_1 - C_{\text{mix}}}$$
Matrix Operations and Systems
Matrices organize data into grids. Addition is element-by-element (same size required); multiplication uses row-column dot products (inner dimensions must match). The determinant and inverse enable solving systems via $\vec{x} = A^{-1}\vec{b}$.
Key Points
- •Dimension rule for multiplication: $(m \times p)(p \times n) = m \times n$ — inner dimensions must match
- •Matrix multiplication is NOT commutative: $AB \neq BA$ in general
- •2×2 determinant: $\det = ad - bc$; if zero, the matrix is singular (no inverse)
- •3×3 determinant: cofactor expansion along any row, alternating signs (+, −, +, ...)
- •Inverse of 2×2: swap the diagonal, negate off-diagonal, divide all by $\det(A)$
- •System solution: $A\vec{x} = \vec{b}$ gives $\vec{x} = A^{-1}\vec{b}$ only when $\det(A) \neq 0$
Formula
$$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc, \quad A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$
Formulas
Difference of Squares
Two perfect squares subtracted factor into a conjugate pair
Formula
$$a^2 - b^2 = (a+b)(a-b)$$
Quadratic Formula
Universal solver for any quadratic equation
Formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Discriminant
Determines the nature of roots without solving
Formula
$$\Delta = b^2 - 4ac$$
Vieta's Formulas
Sum and product of roots from coefficients
Formula
$$\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}$$
Sum and Difference of Cubes
Factorization of $a^3 \pm b^3$ into linear and quadratic factors
Formula
$$a^3 + b^3 = (a+b)(a^2 - ab + b^2), \quad a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$
2×2 Determinant and Inverse
Determinant checks invertibility; inverse solves matrix systems
Formula
$$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc, \quad \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$
Alligation Ratio
Mixing ratio from component values and target mixture value
Formula
$$\frac{Q_1}{Q_2} = \frac{C_{\text{mix}} - C_2}{C_1 - C_{\text{mix}}}$$
Combined Work Time
Time for two workers together from their individual times
Formula
$$T_{\text{together}} = \frac{T_1 \cdot T_2}{T_1 + T_2}$$