https://file.aiquickdraw.com/mathbot/file/1726910849064nmk7ofww.jpg

We are given the area of a trapezium as \( 230 \, \text{cm}^2 \) and its dimensions involving variables \( x \), \( x+2 \), and \( x+6 \). The task is to find the values of \( b \) and \( c \) in the quadratic equation:

\[
2x^2 + bx + c = 0
\]

### Step-by-Step Solution:

The formula for the area of a trapezium is:

\[
\text{Area} = \frac{1}{2} \times ( \text{sum of parallel sides} ) \times \text{height}
\]

- **Sum of parallel sides**: The parallel sides of the trapezium are \( x \) and \( x + 2 \), so their sum is \( x + (x+2) = 2x + 2 \).
- **Height**: The height of the trapezium is given as \( x + 6 \).

Thus, the area formula becomes:

\[
230 = \frac{1}{2} \times (2x + 2) \times (x + 6)
\]

Simplify the right-hand side:

\[
230 = \frac{1}{2} \times (2x^2 + 12x + 2x + 12) = \frac{1}{2} \times (2x^2 + 14x + 12)
\]

Now, multiply everything by 2 to eliminate the fraction:

\[
460 = 2x^2 + 14x + 12
\]

Move everything to one side of the equation:

\[
2x^2 + 14x + 12 - 460 = 0
\]

Simplify:

\[
2x^2 + 14x - 448 = 0
\]

From this equation, we can see that \( b = 14 \) and \( c = -448 \).

### Final Answer:
\[
b = 14, \quad c = -448
\]

Let me know if you'd like any clarifications or more details! 

Here are 5 related questions to explore:
1. How do you derive the formula for the area of a trapezium?
2. How would the equation change if the height was different?
3. What would happen if the area was given as a different value?
4. Can you solve this quadratic equation to find the value of \( x \)?
5. How do the values of \( b \) and \( c \) affect the graph of the quadratic equation?

**Tip:** When solving equations involving geometric figures, always verify that each measurement aligns with the formula you are using to avoid calculation errors.

The area of the trapezium is 230 cm². All measurements are in cm. Show that 2x² + bx + c = 0, where b and c are integers to be determined.

This detailed guide shows how to find the quadratic equation from the area of a trapezium. Given dimensions, we use the trapezium area formula to derive a quadratic equation in the form 2x² + bx + c = 0. Learn how to calculate b and c step-by-step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation