“Algebra: Expressions and Equations”

🟢 Lite — Quick Review (1h–1d)

Rapid summary of algebra for NABTEB mathematics.

Algebra uses letters and symbols to represent numbers and quantities in formulas and equations.

Key Terms:

• Expression: A combination of numbers, variables, and operations (e.g., $3x + 5$). Cannot be solved — no equals sign.
• Equation: A statement that two expressions are equal (e.g., $3x + 5 = 14$). CAN be solved.
• Identity: An equation true for ALL values of the variable (e.g., $(x+1)^2 = x^2 + 2x + 1$)
• Variable: A symbol (usually $x$, $y$, $z$) that represents an unknown quantity

Basic Operations with Algebra:

Operation	Rule	Example
Addition	Like terms only	$3x + 2x = 5x$
Subtraction	Like terms only	$5y - 2y = 3y$
Multiplication	Distribute	$3(x + 4) = 3x + 12$
Division	Divide each term	$(6x + 9)/3 = 2x + 3$

Laws of Indices:

Law	Formula	Example
Product	$a^m \times a^n = a^{m+n}$	$x^3 \times x^4 = x^7$
Quotient	$a^m \div a^n = a^{m-n}$	$y^5 \div y^2 = y^3$
Power of power	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Zero index	$a^0 = 1$	$5^0 = 1$
Negative index	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = \frac{1}{x^2}$
Fractional index	$a^{m/n} = \sqrt[n]{a^m}$	$x^{1/2} = \sqrt{x}$

Solving Linear Equations:

Example: $3x + 7 = 22$

1. $3x = 22 - 7 = 15$
2. $x = 15/3 = 5$

⚡ NABTEB Exam Tip: Always check your answer by substituting back into the original equation. The left-hand side should equal the right-hand side.

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🟡 Standard — Regular Study (2d–2mo)

For NABTEB students who want thorough understanding.

Simultaneous Equations:

Two equations with two unknowns, solved together.

Method 1 — Substitution: Step 1: Express one variable in terms of the other from one equation Step 2: Substitute into the second equation Step 3: Solve for the remaining variable Step 4: Substitute back to find the other variable

Example: $$2x + y = 7 \quad \text{…(1)}$$ $$x - y = 2 \quad \text{…(2)}$$

From (2): $x = y + 2$ Substitute into (1): $2(y+2) + y = 7 \Rightarrow 3y + 4 = 7 \Rightarrow y = 1$ Then $x = 1 + 2 = 3$ Solution: $x = 3, y = 1$

Method 2 — Elimination: Add or subtract equations to eliminate one variable.

Example: $$3x + 2y = 16 \quad \text{…(1)}$$ $$2x - 2y = 4 \quad \text{…(2)}$$

Add (1) + (2): $5x = 20 \Rightarrow x = 4$ Substitute: $3(4) + 2y = 16 \Rightarrow 12 + 2y = 16 \Rightarrow y = 2$

Quadratic Equations:

Equations of the form $ax^2 + bx + c = 0$ (where $a \neq 0$).

Method 1 — Factorisation:

Example: $x^2 - 5x + 6 = 0$ $$(x-2)(x-3) = 0$$ $$x = 2 \text{ or } x = 3$$

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

Method 3 — Completing the Square: $x^2 + bx = -c$ $(x + b/2)^2 - (b/2)^2 = -c$ $(x + b/2)^2 = (b/2)^2 - c$

The Discriminant: $D = b^2 - 4ac$

• $D > 0$: Two distinct real roots
• $D = 0$: Two equal real roots
• $D < 0$: No real roots

⚡ NABTEB Exam Tip: In simultaneous equations with a quadratic, substitution is usually easier. For two linear equations, elimination is often faster. For quadratic equations, try factorisation first — it’s quicker than the formula.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage of algebra for thorough NABTEB preparation.

Partial Fractions:

Expressing a rational function as a sum of simpler fractions.

Case 1 — Linear factors: $$\frac{5x+3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$

Multiply both sides by $(x-1)(x+2)$: $$5x+3 = A(x+2) + B(x-1)$$

Substitute convenient values:

• Let $x = 1$: $5(1)+3 = A(3) + B(0) \Rightarrow 8 = 3A \Rightarrow A = 8/3$
• Let $x = -2$: $5(-2)+3 = A(0) + B(-3) \Rightarrow -7 = -3B \Rightarrow B = 7/3$

$$\frac{5x+3}{(x-1)(x+2)} = \frac{8/3}{x-1} + \frac{7/3}{x+2}$$

Case 2 — Repeated linear factors: $$\frac{2x+1}{(x-3)^2} = \frac{A}{x-3} + \frac{B}{(x-3)^2}$$

Case 3 — Quadratic denominator: $$\frac{x+5}{x^2+1} = \text{cannot be split further (unless numerator has degree ≥ denominator)}$$

Surds:

A surd is an irrational root (e.g., $\sqrt{2}$, $\sqrt{3}$).

Rationalising the Denominator: $$\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{3}}{3}$$

For binomial denominators: $$\frac{5}{2+\sqrt{3}} = \frac{5(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} = \frac{5(2-\sqrt{3})}{4-3} = 5(2-\sqrt{3})$$

Binomial Theorem:

$$(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r}b^r$$

Where $\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Example — $(x+2)^4$: $$= x^4 + 4(2)x^3 + 6(4)x^2 + 4(8)x + 16$$ $$= x^4 + 8x^3 + 24x^2 + 32x + 16$$

Matrices and Determinants:

2×2 Matrix: $$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$$

Determinant: $$\det(A) = ad - bc$$

Inverse (if $\det \neq 0$): $$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}$$

Solving Linear Systems with Matrices:

For $AX = B$: $$X = A^{-1}B \text{ (if } \det(A) \neq 0\text{)}$$

Arithmetic Progression (AP):

$$a, a+d, a+2d, \ldots$$ $$T_n = a + (n-1)d$$ $$S_n = \frac{n}{2}(a + l) = \frac{n}{2}[2a + (n-1)d]$$

Geometric Progression (GP):

$$a, ar, ar^2, \ldots$$ $$T_n = ar^{n-1}$$ $$S_n = \frac{a(1-r^n)}{1-r} \text{ (for } r \neq 1\text{)}$$

If $|r| < 1$, as $n \to \infty$, $S_\infty = \frac{a}{1-r}$

Logarithms:

If $a^x = n$, then $\log_a n = x$

Laws:

• $\log_a(mn) = \log_a m + \log_a n$
• $\log_a(m/n) = \log_a m - \log_a n$
• $\log_a(m^k) = k \log_a m$

Change of base: $$\log_a m = \frac{\log_b m}{\log_b a}$$

Natural logarithm: $\ln x = \log_e x$ where $e ≈ 2.71828$

⚡ NABTEB Quick Reference:

• $a^m \times a^n = a^{m+n}$
• $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
• Discriminant $D = b^2 - 4ac$: $D > 0$ (2 roots), $D = 0$ (equal), $D < 0$ (no real roots)
• $(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r}b^r$
• Determinant: $ad - bc$
• AP: $T_n = a + (n-1)d$; $S_n = \frac{n}{2}(a+l)$
• GP: $T_n = ar^{n-1}$; $S_n = \frac{a(1-r^n)}{1-r}$; $S_\infty = \frac{a}{1-r}$ (for $|r|<1$)
• $\log_a(mn) = \log_a m + \log_a n$