Math Problem Statement

The perimeter of the rectangle is 6x. The dimensions of the rectangle are labeled as 2x and x. (i) Write an expression for the perimeter in terms of x. (ii) Show that x^2 - 4x - 5 = 0.

Solution

The problem shown in the image is about a rectangle. I'll transcribe the problem and solve it step-by-step:

Given Information:

  1. The length of the rectangle is labeled as 2x2x.
  2. The width of the rectangle is labeled as xx.

Part (i) - Expression for Perimeter:

The perimeter of a rectangle is given by the formula: Perimeter=2(length+width)\text{Perimeter} = 2(\text{length} + \text{width}) Substituting the given values: Perimeter=2(2x+x)=2(3x)=6x\text{Perimeter} = 2(2x + x) = 2(3x) = 6x Thus, the perimeter of the rectangle is 6x6x.

Part (ii) - Show that x24x5=0x^2 - 4x - 5 = 0:

The task in this part seems to involve solving for xx using some additional information that relates to the equation x24x5=0x^2 - 4x - 5 = 0. Based on the image, there is likely some connection between the rectangle’s dimensions and this equation. However, the full details of the problem statement in part (ii) are not entirely visible.

To solve the quadratic equation x24x5=0x^2 - 4x - 5 = 0, we can factor it:

We need two numbers that multiply to 5-5 and add to 4-4. These numbers are 5-5 and 11. Thus, we factor the equation as: (x5)(x+1)=0(x - 5)(x + 1) = 0 Solving for xx: x5=0orx+1=0x - 5 = 0 \quad \text{or} \quad x + 1 = 0 x=5orx=1x = 5 \quad \text{or} \quad x = -1

Since xx represents a dimension of a rectangle, we discard the negative solution. Therefore, x=5x = 5.

Conclusion:

The perimeter of the rectangle is 6x6x. When x=5x = 5, the perimeter is 6(5)=306(5) = 30.

If you have further questions, feel free to ask! Would you like a detailed explanation on any part of this problem?

Here are some related questions:

  1. How do we derive the formula for the perimeter of any polygon?
  2. How do you solve more complex quadratic equations using the quadratic formula?
  3. What are other methods for solving quadratic equations besides factoring?
  4. How do the dimensions of a shape affect its area compared to its perimeter?
  5. How can you verify whether the solutions to a quadratic equation are correct?

Tip: When solving quadratic equations by factoring, always verify your factors by expanding the factored form back to the original equation.

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Quadratic Equations

Formulas

Perimeter of a rectangle: P = 2(length + width)
Quadratic equation: ax^2 + bx + c = 0

Theorems

Factoring Quadratics

Suitable Grade Level

Grades 7-9