Algebraic Processes

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your WAEC exam.

Basic Algebraic Laws:

Laws of Indices:

• $a^m \times a^n = a^{m+n}$
• $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^0 = 1$ (for $a \neq 0$)
• $a^{-n} = \frac{1}{a^n}$
• $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Special Indices:

• $a^1 = a$
• $a^{1/2} = \sqrt{a}$
• $a^{1/3} = \sqrt[3]{a}$

Standard Form: Any number can be written as $a \times 10^n$ where $1 \leq a < 10$ Example: $4500 = 4.5 \times 10^3$

Simplifying Expressions:

Example: Simplify $\frac{3x^2 y^3}{12xy^2}$

$$\frac{3}{12} \times x^{2-1} \times y^{3-2} = \frac{1}{4}xy$$

⚡ WAEC Tip: Always check that bases are the same before combining indices. You can only combine $x^2 \times x^3$, not $x^2 \times y^3$.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Factorisation Techniques:

1. Common Factor: $ab + ac = a(b + c)$ Example: $6x^2 + 9x = 3x(2x + 3)$

2. Difference of Two Squares: $a^2 - b^2 = (a + b)(a - b)$ Example: $x^2 - 16 = (x + 4)(x - 4)$

3. Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$ Example: $x^2 + 6x + 9 = (x + 3)^2$

4. Quadratic Trinomials ($ax^2 + bx + c$): Find two numbers that multiply to $ac$ and add to $b$.

Example: $x^2 + 5x + 6$

• $ac = 1 \times 6 = 6$
• Numbers that multiply to 6 and add to 5: 2 and 3
• $x^2 + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 2)(x + 3)$

5. Grouping: $ax + ay + bx + by = (a + b)(x + y)$ Example: $2x + 2y + 3x + 3y = 5x + 5y = 5(x + y)$

Solving Quadratic Equations:

Method 1: Factorisation $x^2 - 5x + 6 = 0$ $(x - 2)(x - 3) = 0$ $x = 2$ or $x = 3$

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

For $ax^2 + bx + c = 0$

Method 3: Completing the Square $x^2 + 6x + 5 = 0$ $x^2 + 6x = -5$ $(x + 3)^2 - 9 = -5$ $(x + 3)^2 = 4$ $x + 3 = \pm 2$ $x = -1$ or $x = -5$

Worked Example: Solve $2x^2 - 5x - 3 = 0$

Using quadratic formula: $a=2, b=-5, c=-3$ $$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)} = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm \sqrt{49}}{4} = \frac{5 \pm 7}{4}$$

$x = \frac{5+7}{4} = 3$ or $x = \frac{5-7}{4} = -\frac{1}{2}$

⚡ Common Mistake: Forgetting that quadratic equations have TWO solutions. Always write both $x = …$ and $x = …$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

Simultaneous Equations:

Substitution Method:

1. Express one variable in terms of the other from one equation
2. Substitute into the second equation
3. Solve for the single variable
4. Substitute back to find the other variable

Worked Example: $2x + y = 7$ … (1) $x - y = 2$ … (2)

From (2): $x = y + 2$ Sub into (1): $2(y + 2) + y = 7$ $2y + 4 + y = 7$ $3y = 3$ $y = 1$

Then $x = 1 + 2 = 3$

Elimination Method: Multiply equations to make coefficients of one variable equal, then add or subtract.

Worked Example: $3x + 2y = 16$ … (1) $5x - 2y = 8$ … (2)

Add (1) + (2): $8x = 24$ $x = 3$

Sub into (1): $9 + 2y = 16$ $2y = 7$ $y = 3.5$

Linear-Quadratic Simultaneous Equations: One linear, one quadratic. Solve linear for one variable, substitute into quadratic.

Algebraic Fractions:

Addition/Subtraction: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

Find LCM of denominators when adding.

Worked Example: $$\frac{3}{x-2} + \frac{2}{x+1}$$

LCM = $(x-2)(x+1)$ $$= \frac{3(x+1) + 2(x-2)}{(x-2)(x+1)} = \frac{3x+3 + 2x-4}{(x-2)(x+1)} = \frac{5x-1}{(x-2)(x+1)}$$

Partial Fractions:

Express $\frac{3x+1}{(x-1)(x+2)}$ in partial fractions.

Assume: $\frac{3x+1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$

$3x+1 = A(x+2) + B(x-1)$ $3x+1 = (A+B)x + (2A - B)$

Equating coefficients: $A + B = 3$ … (1) $2A - B = 1$ … (2)

Adding: $3A = 4$, so $A = \frac{4}{3}$ Then $B = 3 - \frac{4}{3} = \frac{5}{3}$

Answer: $\frac{4/3}{x-1} + \frac{5/3}{x+2}$

The Discriminant: For $ax^2 + bx + c = 0$: $$\Delta = b^2 - 4ac$$

Value of $\Delta$	Nature of Roots
$\Delta > 0$	Two distinct real roots
$\Delta = 0$	Two equal real roots
$\Delta < 0$	No real roots (complex conjugates)

Sum and Product of Roots: For $ax^2 + bx + c = 0$ with roots $\alpha, \beta$: $$\alpha + \beta = -\frac{b}{a}$$ $$\alpha \beta = \frac{c}{a}$$

Forming Equations from Roots: If roots are 2 and 5: $x^2 - (2+5)x + (2 \times 5) = 0$ $x^2 - 7x + 10 = 0$

Logarithms:

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

Logarithm Laws:

• $\log_a (xy) = \log_a x + \log_a y$
• $\log_a \frac{x}{y} = \log_a x - \log_a y$
• $\log_a x^n = n \log_a x$
• $\log_a a = 1$
• $\log_a 1 = 0$

Change of Base: $$\log_a x = \frac{\log_b x}{\log_b a}$$

⚡ WAEC Previous Year Pattern:

Year	Question	Concept
2023	Simplify indices expression	Laws of indices
2022	Solve quadratic equation	Quadratic formula
2021	Simultaneous equations	Elimination method

Exponential Equations:

Example: $2^{x+1} = 32 = 2^5$ Since bases equal: $x + 1 = 5$, so $x = 4$

Example: $3^{2x} = 27 = 3^3$ $2x = 3$, so $x = 1.5$

Sequences and Series:

Arithmetic Sequence: $a, a+d, a+2d, …$ $T_n = a + (n-1)d$ $S_n = \frac{n}{2}(2a + (n-1)d)$

Geometric Sequence: $a, ar, ar^2, …$ $T_n = ar^{n-1}$ $S_n = \frac{a(r^n - 1)}{r - 1}$ (for $r \neq 1$)

⚡ Exam Strategy: In algebra problems, always show your working. For equations, check your answers by substituting back. For simplifying expressions, factor completely before combining.

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