https://file.aiquickdraw.com/mathbot/file/1726004580639c8ik9u1i.jpg

Let's solve the problem step by step based on the image you've provided.

### Problem Breakdown:

1. **The length of the rectangle** is \( (2x - 1) \) cm.
2. **The width of the rectangle** is \( (x + 3) \) cm.

The problem has three parts:

### (i) Write an expression for the area of the rectangle in the form \( ax^2 + bx + c \).

The area \( A \) of a rectangle is given by:
\[
A = \text{Length} \times \text{Width}
\]
Substitute the given expressions for length and width:
\[
A = (2x - 1)(x + 3)
\]
Now, expand the expression:
\[
A = 2x(x + 3) - 1(x + 3)
\]
\[
A = 2x^2 + 6x - x - 3
\]
\[
A = 2x^2 + 5x - 3
\]

Thus, the expression for the area of the rectangle is:
\[
A = 2x^2 + 5x - 3
\]

### (ii) Given that the area of the rectangle is \( 294 \, \text{cm}^2 \), determine the value of \( x \).

We know the area \( A = 294 \, \text{cm}^2 \), so set the expression for the area equal to 294:
\[
2x^2 + 5x - 3 = 294
\]
Now, subtract 294 from both sides:
\[
2x^2 + 5x - 297 = 0
\]
We need to solve this quadratic equation. Let's use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 2x^2 + 5x - 297 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = 5 \)
- \( c = -297 \)

Let's substitute these values into the quadratic formula to find \( x \).

### Calculation:
I'll calculate the discriminant and the values of \( x \).The solutions for \( x \) are:

\[
x_1 = 11 \quad \text{and} \quad x_2 = -13.5
\]

Since \( x \) represents a physical dimension (part of the rectangle's width), it must be positive. Therefore, the valid solution is:

\[
x = 11
\]

### (iii) State the dimensions of the rectangle in centimeters.

Now that we know \( x = 11 \), we can calculate the dimensions of the rectangle.

- **Length**: \( 2x - 1 = 2(11) - 1 = 22 - 1 = 21 \, \text{cm} \)
- **Width**: \( x + 3 = 11 + 3 = 14 \, \text{cm} \)

Thus, the dimensions of the rectangle are:
- **Length**: 21 cm
- **Width**: 14 cm

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can the quadratic formula help solve other geometry problems?
2. Can you solve a similar problem with different expressions for the length and width?
3. How does changing the area affect the quadratic equation?
4. What other methods can be used to solve quadratic equations apart from the quadratic formula?
5. How can this knowledge be applied to more complex 3D shapes like cuboids?

**Tip**: Always check the discriminant before solving a quadratic equation—it tells you whether the solutions are real or complex.

The length of the rectangle is (2x - 1) cm and its width is (x + 3) cm. The area is 294 cm². Solve for x and find the dimensions of the rectangle.

This example involves solving a quadratic equation derived from the area of a rectangle with dimensions (2x - 1) and (x + 3). Given that the area is 294 cm², learn how to find the value of x and the dimensions of the rectangle.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation