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Indices, Quadratics, Inequalities

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

N/A Pathway: N/A

First Principles

Objective: Seamlessly integrate the laws of indices, quadratic factorisation, and algebraic inequalities to model and solve multi-step problems.

1. The Index Ladder (Repeated Multiplication)

Indices quantify exponential scaling. When bases are equal, multiplication aggregates powers: \(a^m \times a^n = a^{m+n}\), while division removes factors: \(\frac{a^m}{a^n} = a^{m-n}\). A power of a power multiplies dimensions: \((a^m)^n = a^{mn}\).

2. Quadratic Geometry (Parabolic Shapes)

A quadratic equation \(ax^2 + bx + c = 0\) represents where a parabola intersects the horizontal axis. Factorising into \((x-p)(x-q)=0\) relies on the Zero Product Property: if a product is zero, at least one factor must be zero, yielding critical boundaries at \(x = p\) and \(x = q\).

3. Sign Analysis on Inequalities

An inequality such as \(ax^2 + bx + c > 0\) identifies the domain where the parabola lies strictly above the \(x\)-axis. For \(a > 0\) (u-shaped), the function is positive on the outer tails (\(x < p\) or \(x > q\)) and negative between the roots (\(p < x < q\)).

Key Formulas

\[ a^m \times a^n = a^{m+n} \] — Product Law: Multiply powers of the same base by adding exponents.
\[ \frac{a^m}{a^n} = a^{m-n} \] — Quotient Law: Divide powers of the same base by subtracting exponents.
\[ (a^m)^n = a^{m \cdot n} \] — Power Law: Raise a power to an exponent by multiplying exponents.
\[ a^{-n} = \frac{1}{a^n}, \quad a^{\frac{1}{n}} = \sqrt[n]{a}, \quad a^0 = 1 \text{ (for } a \neq 0\text{)} \] — Negative, Fractional & Zero Powers: Core reciprocal and radical equivalences.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] — The Quadratic Formula: Solves any general quadratic equation \(ax^2 + bx + c = 0\).
\[ \Delta = b^2 - 4ac \] — The Discriminant: \(\Delta > 0\) gives 2 distinct real roots, \(\Delta = 0\) gives 1 repeated root, \(\Delta < 0\) yields no real roots.
\[ (x - p)(x - q) < 0 \implies p < x < q \quad (\text{assuming } p < q) \] — Inside Region: Upward parabola below the \(x\)-axis.
\[ (x - p)(x - q) > 0 \implies x < p \quad \text{or} \quad x > q \quad (\text{assuming } p < q) \] — Outside Region: Upward parabola above the \(x\)-axis.

Worked Examples

Example 1 (Easy) — Simplifying Combined Index Expressions:
Problem: Simplify \(\frac{2^3 \times 2^4}{2^5}\) completely to an integer.
  1. Combine the numerator using the Product Law:
    \[ 2^3 \times 2^4 = 2^{3+4} = 2^7 \]
  2. Apply the Quotient Law to divide:
    \[ \frac{2^7}{2^5} = 2^{7-5} = 2^2 \]
  3. Evaluate:
    \[ 2^2 = 4 \]
Example 2 (Medium) — Solving Area Geometry via Factorisation:
Problem: A farmer in Kilifi constructs a rectangular goat pen. The length is \(x + 5\) metres and the width is \(x - 2\) metres. If the total area is \(30\text{ m}^2\), calculate the value of \(x\).
  1. Set up the area formula: \(\text{Area} = \text{Length} \times \text{Width}\)
    \[ (x + 5)(x - 2) = 30 \]
  2. Expand the quadratic expression:
    \[ x^2 - 2x + 5x - 10 = 30 \implies x^2 + 3x - 10 = 30 \]
  3. Set the quadratic equation to zero:
    \[ x^2 + 3x - 40 = 0 \]
  4. Factorise the quadratic: Find factors of \(-40\) that sum to \(+3\) \(\to +8\) and \(-5\):
    \[ (x + 8)(x - 5) = 0 \]
  5. Solve for \(x\): \(x = -8\) or \(x = 5\). Since physical length cannot be negative (\(x - 2 > 0\)), discard \(x = -8\).
    Solution: \(x = 5\).
Example 3 (Hard) — Solving an Index Equation with Base Conversion:
Problem: Solve for \(x\): \(3^{2x} = 9 \times 27^{x}\).
  1. Express every term with base 3: Notice that \(9 = 3^2\) and \(27 = 3^3\).
    \[ 3^{2x} = 3^2 \times (3^3)^x \]
  2. Apply the Power of a Power Law: \((3^3)^x = 3^{3x}\)
    \[ 3^{2x} = 3^2 \times 3^{3x} \]
  3. Combine exponents on the right side:
    \[ 3^{2x} = 3^{2 + 3x} \]
  4. Equate the exponents (since bases are equal):
    \[ 2x = 2 + 3x \]
  5. Rearrange to solve for \(x\):
    \[ 2x - 3x = 2 \implies -x = 2 \implies x = -2 \]

Common Mistakes

Mistake Adding exponents when terms are added, e.g., writing \(2^3 + 2^4 = 2^7\) or \(a^{m-n} = a^m - a^n\).
Correction Index laws apply only to multiplication and division across identical bases. For addition, calculate term values directly: \(2^3 + 2^4 = 8 + 16 = 24\) (whereas \(2^7 = 128\)).
Mistake Forgetting to reverse the inequality sign when dividing or multiplying by a negative number.
Correction If \(-2x < 6\), dividing both sides by \(-2\) gives \(x > -3\). Multiplying or dividing by a negative reflects values across zero on the number line, flipping the ordering.
Mistake Stating quadratic inequality solutions as a single combined inequality like \(5 < x < -1\).
Correction Disjoint regions cannot be joined by "and". An upward parabola above zero requires two separate inequalities: \(x < -1 \text{ or } x > 5\).
Mistake Cancelling \(x\) from both sides of \(x^2 = 5x\) to obtain only \(x = 5\).
Correction Dividing by \(x\) illegally eliminates the root \(x = 0\). Always rearrange to \(x^2 - 5x = 0 \implies x(x - 5) = 0\) giving \(x = 0\) or \(x = 5\).

Real World

Index Growth Component: At tier \(n = 3\), data cost is \(50 \times 2^3 = 50 \times 8 = 400\text{ KES}\).
Quadratic Discount: The quadratic term \(n^2 - 20n = 3^2 - 20(3) = 9 - 60 = -51\text{ KES}\) discount.
Inequality Budget Check: Net cost at tier 3 is \(400 - 51 = 349\text{ KES}\). Since \(349 \leq 400\), tier 3 is sustainable. At tier 4: \(50 \times 2^4 + (16 - 80) = 800 - 64 = 736\text{ KES} > 400\text{ KES}\), exceeding the budget threshold.

Practice

Simplify the expression: \[ \frac{2^3 \times 2^4}{2^5} \] (Type only the number, e.g., 7)
Review the concepts above.
Simplify the expression to a single simplified fraction: \[ \frac{2^3 \times 5^2}{2^1 \times 5^3} \] (Type only the fraction, e.g., 2/3)
Review the concepts above.
Solve for \(x\): \[ 3^{2x} = 9 \times 3^x \] (Type only the number, e.g., 5)
Review the concepts above.
Omar is designing a rectangular garden in Kilifi. The length is \((x + 5)\) metres and the width is \((x - 2)\) metres. If the area of the garden is \(30\text{ m}^2\), what is the positive value of \(x\)? (Type only the number, e.g., 42)
Review the concepts above.
Solve for \(x\): \[ 3^{2x} = 9 \times 27^{x} \] (Type only the number, e.g., -5)
Review the concepts above.
How many integer values of \(x\) satisfy the strict quadratic inequality: \[ 3x^2 - 12x + 9 < 0 \] (Type only the number, e.g., 5)
Review the concepts above.