The length and breadth of a rectangle are (-3x + 4) and (x - 11), respectively. Find the area of the rectangle.
Understand the Problem
The question provides the expressions for the length and breadth of a rectangle in terms of 'x'. It asks us to find the area of the rectangle. To do so, we need to multiply the given expressions for length and breadth and simplify the resulting expression.
Answer
$6x^2 + 5x - 6$
Answer for screen readers
The area of the rectangle is $6x^2 + 5x - 6$.
Steps to Solve
- Write the formula for the area of a rectangle
The area of a rectangle is given by the formula: $$ \text{Area} = \text{length} \times \text{breadth} $$
- Substitute the given expressions for length and breadth
Given that length = $(2x + 3)$ and breadth = $(3x - 2)$, substitute these into the area formula: $$ \text{Area} = (2x + 3)(3x - 2) $$
- Expand the expression
Expand the product of the two binomials using the distributive property (also known as FOIL method): $$ \text{Area} = 2x(3x) + 2x(-2) + 3(3x) + 3(-2) $$
- Simplify each term
Multiply each term: $$ \text{Area} = 6x^2 - 4x + 9x - 6 $$
- Combine like terms
Combine the 'x' terms: $$ \text{Area} = 6x^2 + 5x - 6 $$
The area of the rectangle is $6x^2 + 5x - 6$.
More Information
The area is expressed as a quadratic expression in terms of $x$. If the value of $x$ is known, we can substitute $x$ into the expression to find the numerical value of the area.
Tips
A common mistake is to incorrectly apply the distributive property when expanding the product of the two binomials. Ensure each term in the first binomial is multiplied by each term in the second binomial. Another common mistake is to make a sign error when multiplying. For example, incorrectly calculating $2x \times -2$ as $4x$ instead of $-4x$.
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