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The diagram shows a rectangle.the area of the rectangle is 20 cm²     What is the value Of X,in centimetres. 

To solve for the value of \( x \) in the given rectangle, we'll use the information provided:

- The area of the rectangle is given as \( 20 \, \text{cm}^2 \).
- The length of the rectangle is \( x + 4 \) cm.
- The width of the rectangle is \( x - 1 \) cm.

The formula for the area of a rectangle is:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

Substituting the given expressions:

\[
(x + 4)(x - 1) = 20
\]

Now, we need to solve this quadratic equation for \( x \):

1. Expand the left side:

\[
x^2 - x + 4x - 4 = 20
\]

\[
x^2 + 3x - 4 = 20
\]

2. Bring all terms to one side of the equation:

\[
x^2 + 3x - 24 = 0
\]

3. Factor the quadratic equation:

\[
(x + 6)(x - 4) = 0
\]

4. Set each factor to zero and solve for \( x \):

\[
x + 6 = 0 \quad \text{or} \quad x - 4 = 0
\]

\[
x = -6 \quad \text{or} \quad x = 4
\]

Since \( x \) represents a length, it must be a positive value. Therefore, \( x = 4 \) cm.

**Conclusion**: The value of \( x \) is \( 4 \) cm.

Would you like any further details, or do you have any additional questions?

### Follow-up Questions:
1. What would the dimensions of the rectangle be with \( x = 4 \) cm?
2. How would the solution change if the area were different?
3. Could we solve this equation graphically, and what would that involve?
4. How do you determine if a quadratic equation has real or complex roots?
5. What would happen if both solutions for \( x \) were negative?
6. How can we use the quadratic formula to solve the equation instead of factoring?
7. How would you check if the factored solution is correct?
8. What is the significance of the discriminant in a quadratic equation?

**Tip**: Always check if your solutions make sense in the context of the problem—especially in geometry, where dimensions should be positive.

Learn how to find the length of a rectangle when given its area of 20 cm². This problem involves applying the formula for the area of a rectangle and solving a quadratic equation. Suitable for students in Grades 7-9.

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