calculate the perimeter of a rectangle whose area is 25x^2 - 35x + 12

To calculate the perimeter of a rectangle, we need to know the length and width of the rectangle. However, you've provided the **area** as a quadratic expression: 

\[
\text{Area} = 25x^2 - 35x + 12
\]

Assume that the area \( A = \text{length} \times \text{width} \). We need to factor the quadratic expression to find the dimensions (length and width) of the rectangle.

### Step 1: Factor the quadratic expression

We want to factor \( 25x^2 - 35x + 12 \).

First, multiply the coefficient of \( x^2 \) (which is 25) by the constant term (12):
\[
25 \times 12 = 300
\]
Now, we look for two numbers that multiply to 300 and add up to -35 (the coefficient of \( x \)). These numbers are -15 and -20.

So, we rewrite the middle term:
\[
25x^2 - 15x - 20x + 12
\]

Now, factor by grouping:
\[
(25x^2 - 15x) - (20x - 12) = 5x(5x - 3) - 4(5x - 3)
\]
\[
= (5x - 4)(5x - 3)
\]

Thus, the area factors as:
\[
\text{Area} = (5x - 4)(5x - 3)
\]

### Step 2: Perimeter formula

The perimeter \( P \) of a rectangle is given by:
\[
P = 2 \times (\text{length} + \text{width})
\]

From the factorization, we can assign:
\[
\text{Length} = 5x - 4, \quad \text{Width} = 5x - 3
\]

Now, substitute these into the perimeter formula:
\[
P = 2 \times \left((5x - 4) + (5x - 3)\right)
\]
\[
P = 2 \times (10x - 7)
\]
\[
P = 20x - 14
\]

### Final Answer:
The perimeter of the rectangle is:
\[
P = 20x - 14
\]

Would you like further details on this calculation or have any questions?

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Here are some related questions:

1. How do you factor more complex quadratic expressions?
2. What is the general formula for the area and perimeter of rectangles?
3. How can you solve quadratic equations when factoring is difficult?
4. What happens if the quadratic expression cannot be factored easily?
5. Can the dimensions of a rectangle ever be negative?

**Tip:** When factoring quadratic expressions, always check your factored terms by multiplying them back to ensure they produce the original expression.

Learn how to calculate the perimeter of a rectangle whose area is given by the quadratic expression 25x^2 - 35x + 12. This solution includes factoring the quadratic expression to find the dimensions and applying the perimeter formula. Suitable for students in Grades 9-10.

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