What is a Binary Operation?

A binary operation is a rule for combining two quantities to produce a unique result. Unlike ordinary operations (+, −, ×, ÷), the symbol used in a binary operation does not have a standard meaning—it is defined differently in each problem.

Common symbols include *, Δ, ⊗, and ⊕. The operation is defined by a formula given in the problem.

How to Evaluate a Binary Operation

When you see an expression like $a * b = 5a - b$, follow these steps:

1. Identify the operation symbol (*, Δ, etc.)
2. Substitute the given values into the defined formula
3. Simplify the result

Worked Example

If $a * b = 3a + b$, find the value of $5 * 8$.

Solution

The operation is defined as $a * b = 3a + b$.

Substitute $a = 5$ and $b = 8$:

$5 * 8 = 3(5) + 8 = 15 + 8 = 23$

Therefore, $5 * 8 = 23$.

Worked Example with Nested Operations

If $x * y = x + 3y$, find the value of $5 * (8 * 6)$.

Solution

First, evaluate the inner operation $8 * 6$:

$8 * 6 = 8 + 3(6) = 8 + 18 = 26$

Now evaluate $5 * 26$:

$5 * 26 = 5 + 3(26) = 5 + 78 = 83$

Therefore, $5 * (8 * 6) = 83$.

Definition

A quadratic expression is an algebraic expression where the highest power (degree) of the variable is 2. In general form:

$ax^2 + bx + c$

where $a \neq 0$, and $a$, $b$, $c$ are real numbers.

• $ax^2$ is the quadratic term (coefficient is $a$)
• $bx$ is the linear term (coefficient is $b$)
• $c$ is the constant term

Expanding Quadratic Expressions

To expand an expression like $(2y + 3)(y + 1)$, multiply each term in the first bracket by each term in the second bracket:

$(2y + 3)(y + 1) = 2y(y + 1) + 3(y + 1)$ $= 2y^2 + 2y + 3y + 3$ $= 2y^2 + 5y + 3$

Key Identities

These three identities are useful for expanding quadratic expressions:

1. $(a + b)^2 = a^2 + 2ab + b^2$
2. $a^2 - b^2 = (a + b)(a - b)$
3. $(x + a)(x + b) = x^2 + (a + b)x + ab$

What is a Quadratic Equation?

A quadratic equation is formed when a quadratic expression equals zero:

$ax^2 + bx + c = 0$

The solutions are called roots or zeros.

Zero Factor Theorem

If $(x + a)(x + b) = 0$, then either $x + a = 0$ or $x + b = 0$ (or both). This means at least one factor must equal zero.

Steps to Solve by Factorization

1. Write the equation in standard form: $ax^2 + bx + c = 0$
2. Factorize the quadratic expression into two linear factors
3. Set each factor equal to zero
4. Solve for the variable

Worked Example

Solve $x^2 - 2x - 24 = 0$

Solution

Step 1: The equation is already in standard form.

Step 2: Factorize. Find two numbers that multiply to $-24$ and add to $-2$.

These numbers are $-6$ and $+4$:

$x^2 - 2x - 24 = x^2 - 6x + 4x - 24$ $= x(x - 6) + 4(x - 6)$ $= (x - 6)(x + 4)$

Step 3: Set each factor to zero:

$x - 6 = 0 \quad \text{or} \quad x + 4 = 0$

Step 4: Solve:

$x = 6 \quad \text{or} \quad x = -4$

Worked Example from Real Life

A rectangular garden is 6 metres wide and 8 metres long. What length should be added to the shorter side and reduced from the longer side to form a rectangular garden with an area of 45 square metres?

Solution

Let $x$ be the length added to the shorter side (and reduced from the longer side).

New dimensions:

• Width = $(6 + x)$ m
• Length = $(8 - x)$ m

Area equation:

$(6 + x)(8 - x) = 45$

Expand and simplify:

$48 + 8x - 6x - x^2 = 45$ $x^2 - 2x - 3 = 0$ $(x - 3)(x + 1) = 0$

Therefore, $x = 3$ or $x = -1$.

Since length cannot be negative, $x = 3$ metres should be added to the shorter side.

When factorization is not possible, we use completing the square or the quadratic formula.

Completing the Square

To make $x^2 + bx$ a perfect square, add $(\frac{b}{2})^2$.

What must be added to $x^2 + 10x$ to make it a perfect square?

Half of 10 is 5, and $5^2 = 25$. Therefore, add 25:

$x^2 + 10x + 25 = (x + 5)^2$

Quadratic Formula

For $ax^2 + bx + c = 0$ where $a \neq 0$:

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

This formula works for all quadratic equations.

• Binary operations use defined symbols to combine two values
• Quadratic expressions have the form $ax^2 + bx + c$ where the highest power is 2
• Quadratic equations are solved by factorization, completing the square, or using the quadratic formula
• Always check your solutions, especially in real-world problems

In Tanzania, quadratic equations are used in everyday business situations. For example, when a shopkeeper at a local market wants to find the price per mango that maximizes profit, or when a farmer needs to determine the dimensions of a rectangular garden to plant a specific area of vegetables, these algebraic methods help calculate the exact values needed. A vendor at Mwinyi Mkuu market in Zanzibar might use such calculations when budgeting for stock or calculating profit margins on imported goods.