Q2(i):

Represent the following situations in the form of quadratic equations : (i) The area of a rectangular plot is $528$ $m^2$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Given:

1. The shape of the plot is rectangular.
2. The area of the rectangular plot ($A$) = $528$ $m^2$.
3. The relationship between length ($l$) and breadth ($b$): The length is one more than twice its breadth.

To Find:

Represent the given situation in the form of a quadratic equation.

Step 1: Defining the Variables

Let the breadth of the rectangular plot be $x$ metres.
According to the problem, the length is one more than twice the breadth.
Therefore, the length of the plot $l = (2x + 1)$ metres.

Step 2: Applying the Formula for Area

The area of a rectangle is given by the formula:
$Area = Length \times Breadth$
$A = l \times b$

Step 3: Formulating the Equation

Substitute the given values and the expressions defined in Step 1 into the area formula:
$528 = (2x + 1) \times x$

Step 4: Simplifying the Expression

Distribute $x$ into the parentheses:
$528 = 2x^2 + x$

Step 5: Rearranging into Standard Quadratic Form

The standard form of a quadratic equation is $ax^2 + bx + c = 0$.
Subtract $528$ from both sides of the equation:
$2x^2 + x - 528 = 0$

Justification:

The equation $2x^2 + x - 528 = 0$ is a quadratic equation because it is a polynomial equation of degree 2, where $a = 2$, $b = 1$, and $c = -528$.

Final Answer: The quadratic equation representing the situation is $2x^2 + x - 528 = 0$, where $x$ represents the breadth of the plot in metres.