the area of a rectangular of paper is (x² + 7x - 30) sq.cm. How long is the piece of paper if it is ( x - 3 ) cm wide?

To determine the length of the rectangular piece of paper, we can use the formula for the area of a rectangle:

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

Given that the area of the rectangle is \( x^2 + 7x - 30 \) square centimeters and the width is \( x - 3 \) centimeters, we can find the length by dividing the area by the width:

\[
\text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{x^2 + 7x - 30}{x - 3}
\]

Now, let's simplify this expression:

1. **Factor the numerator** \( x^2 + 7x - 30 \):

   We need to factor \( x^2 + 7x - 30 \). We look for two numbers that multiply to \(-30\) and add to \(7\). These numbers are \(10\) and \(-3\).

   So, we can factor the quadratic expression as:

   \[
   x^2 + 7x - 30 = (x + 10)(x - 3)
   \]

2. **Substitute the factorization into the length formula**:

   \[
   \text{Length} = \frac{(x + 10)(x - 3)}{x - 3}
   \]

3. **Cancel the common factor** \( x - 3 \):

   \[
   \text{Length} = x + 10
   \]

Therefore, the length of the rectangular piece of paper is \( x + 10 \) centimeters.

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Would you like further details or have any other questions?

Here are five related questions you might find interesting:

1. What would be the length if \( x = 5 \) cm?
2. How would you find the length if the area were \( x^2 + 9x + 20 \) sq.cm?
3. Can you verify the factorization of \( x^2 + 7x - 30 \)?
4. What would the length be if the width was \( x + 2 \) cm instead?
5. How does the length change as \( x \) increases?

**Tip:** When working with quadratic expressions, factoring can often simplify the problem and help you find the solution more easily.

Learn how to calculate the length of a rectangular piece of paper given its area formula (x² + 7x - 30) sq.cm and width (x - 3) cm. This problem involves algebraic manipulation, factoring quadratic expressions, and applying the formula for the area of a rectangle. Suitable for students in Grades 9-12.

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