Math Problem Statement
calculate the perimeter of a rectangle whose area is 25x^2 - 35x + 12
Solution
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:
Assume that the area . 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 .
First, multiply the coefficient of (which is 25) by the constant term (12): Now, we look for two numbers that multiply to 300 and add up to -35 (the coefficient of ). These numbers are -15 and -20.
So, we rewrite the middle term:
Now, factor by grouping:
Thus, the area factors as:
Step 2: Perimeter formula
The perimeter of a rectangle is given by:
From the factorization, we can assign:
Now, substitute these into the perimeter formula:
Final Answer:
The perimeter of the rectangle is:
Would you like further details on this calculation or have any questions?
Here are some related questions:
- How do you factor more complex quadratic expressions?
- What is the general formula for the area and perimeter of rectangles?
- How can you solve quadratic equations when factoring is difficult?
- What happens if the quadratic expression cannot be factored easily?
- 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.
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Math Problem Analysis
Mathematical Concepts
Algebra
Factoring Quadratic Expressions
Geometry
Rectangular Perimeter and Area
Formulas
Area of rectangle = length × width
Perimeter of rectangle = 2 × (length + width)
Theorems
Factoring quadratic expressions
Distributive property
Suitable Grade Level
Grades 9-10
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