The length and it's width of the rectangular garden measure 7m and 4m, respectively. If it's length is increased by a number in meter and width is increased by thrice of the same number. Representthe equation if the new area is 90 square meter find the quadratic equation?

Given the problem:

- Original length of the rectangular garden \( l = 7 \) meters
- Original width of the rectangular garden \( w = 4 \) meters
- Length increased by \( x \) meters, making the new length \( l' = 7 + x \) meters
- Width increased by \( 3x \) meters, making the new width \( w' = 4 + 3x \) meters
- New area of the garden \( A = 90 \) square meters

We need to form and solve the quadratic equation based on this information.

### Step 1: Express the new area in terms of \( x \).

The new area of the rectangular garden is given by:

\[
A = l' \times w'
\]

Substituting the values:

\[
90 = (7 + x)(4 + 3x)
\]

### Step 2: Expand the equation.

Expand the right side of the equation:

\[
90 = 7 \times 4 + 7 \times 3x + x \times 4 + x \times 3x
\]

\[
90 = 28 + 21x + 4x + 3x^2
\]

Combine like terms:

\[
90 = 3x^2 + 25x + 28
\]

### Step 3: Rearrange the equation into standard quadratic form.

Move all terms to one side of the equation:

\[
3x^2 + 25x + 28 - 90 = 0
\]

\[
3x^2 + 25x - 62 = 0
\]

This is the required quadratic equation:

\[
\boxed{3x^2 + 25x - 62 = 0}
\]

Would you like to see the steps for solving this quadratic equation or need any further clarifications?

Here are some related questions to explore:
1. How do you solve a quadratic equation using the quadratic formula?
2. What are the possible values of \( x \) for the given quadratic equation?
3. How does changing the length and width affect the area of a rectangle?
4. Can you find the roots of the quadratic equation \( 3x^2 + 25x - 62 = 0 \)?
5. What are the applications of quadratic equations in real-life problems like this one?

**Tip:** When forming quadratic equations from word problems, always identify the variables and express the known relationships in terms of these variables.

Explore how to find the quadratic equation representing the area of a rectangular garden with given dimensions and increases. This problem involves algebraic manipulation and the quadratic formula theorem. Suitable for students in Grades 9-12.

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