4.2 Basic Algebra, Linear Equations & Exponents
Key Takeaways
- For any quadratic equation ax^2 + bx + c = 0, the roots are given by x = (-b ± sqrt(b^2 - 4ac)) / (2a), where D = b^2 - 4ac is the discriminant.
- Key standard algebraic identities include (a+b)^2 = a^2 + 2ab + b^2, (a-b)^2 = a^2 - 2ab + b^2, and (a+b)(a-b) = a^2 - b^2.
- Product and quotient laws of exponents state that a^m * a^n = a^(m+n) and a^m / a^n = a^(m-n) for any non-zero base a.
- Power and negative exponent rules state that (a^m)^n = a^(m*n), a^(-n) = 1 / (a^n), and a^0 = 1 (for a ≠ 0).
- A system of two linear equations a1*x + b1*y = c1 and a2*x + b2*y = c2 has a unique solution if the determinant D = a1*b2 - a2*b1 ≠ 0.
Last updated: July 2026
Algebraic Expressions, Factoring & Quadratic Equations
Algebra forms the structural foundation of technical calculation in the PAF Airman selection process. Candidates must master algebraic expansion, factorization, linear systems, and exponent laws.
Standard Algebraic Identities
Polynomial expressions can often be simplified using classic identity expansions:
(a + b)^2 &= a^2 + 2ab + b^2 \\ (a - b)^2 &= a^2 - 2ab + b^2 \\ a^2 - b^2 &= (a + b)(a - b) \\ (a + b)^3 &= a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a + b) \\ a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \\ a^3 - b^3 &= (a - b)(a^2 + ab + b^2) \end{aligned}$$ | Identity Name | Formula | Common Application | | :--- | :--- | :--- | | **Difference of Squares** | $a^2 - b^2 = (a+b)(a-b)$ | Fast mental multiplication (e.g., $43 \times 37 = (40+3)(40-3)$) | | **Square of Sum** | $(a+b)^2 = a^2 + 2ab + b^2$ | Expanding binomially, completing the square | | **Square of Difference** | $(a-b)^2 = a^2 - 2ab + b^2$ | Simplifying radical expressions | ### Worked Example: Factoring Quadratic Expressions **Problem:** Solve the quadratic equation $2x^2 - 7x + 6 = 0$ by factoring. **Solution:** 1. Find two numbers that multiply to give $a \cdot c = 2 \times 6 = 12$ and add up to $b = -7$. 2. The two numbers are $-3$ and $-4$ (since $(-3) \times (-4) = 12$ and $(-3) + (-4) = -7$). 3. Rewrite the middle term: $2x^2 - 4x - 3x + 6 = 0$. 4. Group terms: $2x(x - 2) - 3(x - 2) = 0 \implies (2x - 3)(x - 2) = 0$. 5. Set each factor to zero: - $2x - 3 = 0 \implies x = \frac{3}{2} = 1.5$ - $x - 2 = 0 \implies x = 2$ **Roots:** $x = 1.5$ or $x = 2$. ## The Quadratic Formula & Discriminant For any quadratic equation $ax^2 + bx + c = 0$ ($a \neq 0$): $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The term under the square root, **$D = b^2 - 4ac$**, is called the **Discriminant**: - **$D > 0$**: Two distinct real roots. - **$D = 0$**: Two equal real roots (one repeated root). - **$D < 0$**: No real roots (two complex conjugate roots).Linear Systems & Laws of Exponents
Systems of Simultaneous Linear Equations
A system of two linear equations in two variables takes the form:
Methods of Solution
- Elimination Method: Multiply one or both equations by constants so that coefficients of one variable match, then add or subtract equations.
- Substitution Method: Express one variable in terms of the other from one equation and substitute it into the second equation.
- Cramer's Rule (Determinants): where $D = a_1 b_2 - a_2 b_1$, $D_x = c_1 b_2 - c_2 b_1$, and $D_y = a_1 c_2 - a_2 c_1$. (Valid when $D \neq 0$).
Worked Example: System of Linear Equations
Problem: Solve the system:
Solution (Elimination Method):
- Multiply the second equation by $2$ to match the coefficient of $y$:
- Add this modified equation to the first equation:
- Substitute $x = 2$ into $3x + 4y = 18$:
Solution Pair: $(x, y) = (2, 3)$.
Laws of Exponents and Indices
Exponent questions in PAF Airman examinations test algebraic simplification under exponential rules.
| Rule Name | Mathematical Law | Example |
|---|---|---|
| Product Rule | $a^m \cdot a^n = a^{m+n}$ | $2^3 \cdot 2^4 = 2^7 = 128$ |
| Quotient Rule | $\frac{a^m}{a^n} = a^{m-n}$ | $\frac{5^6}{5^2} = 5^4 = 625$ |
| Power of Power | $(a^m)^n = a^{m \cdot n}$ | $(3^2)^3 = 3^6 = 729$ |
| Power of Product | $(a \cdot b)^n = a^n \cdot b^n$ | $(2 \cdot 5)^3 = 2^3 \cdot 5^3 = 1000$ |
| Negative Exponent | $a^{-n} = \frac{1}{a^n}$ | $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ |
| Fractional Exponent | $a^{m/n} = \sqrt[n]{a^m}$ | $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$ |
| Zero Exponent | $a^0 = 1 \quad (a \neq 0)$ | $99^0 = 1$ |
Worked Exponential Equation
Problem: Solve for $x$ in the equation: $8^{x-1} = 32^{x-3}$.
Solution:
- Express both sides with a common base of $2$:
- Equate exponents:
Result: $x = 6$.
Test Your Knowledge
If 4^(2x - 1) = 64, what is the value of x?
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Test Your Knowledge
What are the roots of the quadratic equation x^2 - 9x + 20 = 0?
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Test Your Knowledge
If 2x + 3y = 12 and x - y = 1, what is the value of y?
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