Q3:

Find two numbers whose sum is $27$ and product is $182$.

Given:

1. The sum of two numbers is $27$.

2. The product of the same two numbers is $182$.

To Find:

The two numbers.

Step 1: Defining the Variables

Let the first number be $x$.

Since the sum of the two numbers is $27$, the second number can be expressed as $(27 - x)$.

Step 2: Formulating the Quadratic Equation

According to the problem, the product of these two numbers is $182$. Therefore, we can set up the following equation:

$x(27 - x) = 182$

Step 3: Simplifying the Equation

Distribute $x$ into the parentheses:

$27x - x^2 = 182$

Rearrange the terms to form a standard quadratic equation of the form $ax^2 + bx + c = 0$:

$-x^2 + 27x - 182 = 0$

Multiply the entire equation by $-1$ to make the leading coefficient positive:

$x^2 - 27x + 182 = 0$

Step 4: Solving by Factorization (Splitting the Middle Term)

We need to find two numbers that multiply to $182$ and add up to $-27$.

Prime factorization of $182$:

$182 = 2 \times 91 = 2 \times 7 \times 13 = 14 \times 13$

Since the sum must be $-27$, we choose $-14$ and $-13$ because $(-14) + (-13) = -27$ and $(-14) \times (-13) = 182$.

Rewrite the middle term:

$x^2 - 14x - 13x + 182 = 0$

Step 5: Grouping and Factoring

Group the terms:

$(x^2 - 14x) - (13x - 182) = 0$

Factor out the common terms from each group:

$x(x - 14) - 13(x - 14) = 0$

Factor out the common binomial $(x - 14)$:

$(x - 14)(x - 13) = 0$

Step 6: Finding the Roots

Set each factor to zero [Using the Zero Product Property]:

Case 1: $x - 14 = 0 \implies x = 14$

Case 2: $x - 13 = 0 \implies x = 13$

If the first number is $14$, the second number is $27 - 14 = 13$.

If the first number is $13$, the second number is $27 - 13 = 14$.

Final Answer: The two numbers are 13 and 14.