Q2(iii):

Represent the following situations in the form of quadratic equations : (iii) Rohan’s mother is $26$ years older than him. The product of their ages (in years) $3$ years from now will be $360$. We would like to find Rohan’s present age.

Given:

1. Rohan’s mother is $26$ years older than Rohan.

2. The product of their ages $3$ years from now will be $360$.

To Find:

Represent the given situation in the form of a quadratic equation in terms of Rohan's present age.

Step 1: Defining the Variables

Let the present age of Rohan be $x$ years.

Since Rohan’s mother is $26$ years older than him, the present age of Rohan’s mother is $(x + 26)$ years.

Step 2: Determining Ages 3 Years from Now

After $3$ years, the age of each person will increase by $3$ years.

Rohan’s age after $3$ years = $(x + 3)$ years.

Rohan’s mother’s age after $3$ years = $(x + 26) + 3 = (x + 29)$ years.

Step 3: Formulating the Equation

According to the problem, the product of their ages $3$ years from now is $360$.

Therefore, we set up the equation:

$(x + 3)(x + 29) = 360$

Step 4: Expanding and Simplifying the Equation

Using the distributive property (FOIL method) to expand the left side:

$x(x + 29) + 3(x + 29) = 360$

$x^2 + 29x + 3x + 87 = 360$

[Combining like terms $29x$ and $3x$]:

$x^2 + 32x + 87 = 360$

Step 5: Bringing the Equation to Standard Form

The standard form of a quadratic equation is $ax^2 + bx + c = 0$. Subtract $360$ from both sides:

$x^2 + 32x + 87 - 360 = 0$

$x^2 + 32x - 273 = 0$

Justification:

The resulting equation $x^2 + 32x - 273 = 0$ is a polynomial of degree $2$, which satisfies the definition of a quadratic equation where $a=1$, $b=32$, and $c=-273$.

Final Answer: The quadratic equation representing the situation is $x^2 + 32x - 273 = 0$, where $x$ is Rohan's present age.