UpStudy Free Solution:

To form a quadratic equation based on the given information, we need to equate the areas of the square \(ABCD\) and the right-angled triangle \(PQR\).

Area of the Square \(ABCD\):
The side length of the square is given as \(3x + 2\). The area of the square is:
\[\text { Area of square} = ( \text { side length} ) ^ 2 = ( 3x + 2) ^ 2\]

Area of the Right-Angled Triangle \(PQR\):
The base of the triangle \(PQR\) is \(3x + 2\) and the height is \(4\). The area of the triangle is:
\[\text { Area of triangle} = \frac { 1} { 2} \times \text { base} \times \text { height} = \frac { 1} { 2} \times ( 3x + 2) \times 4\]
Simplifying the area of the triangle:
\[\text { Area of triangle} = 2 \times ( 3x + 2) = 6x + 4\]
Since the areas of the square and the triangle are equal, we set the two expressions equal to each other:
\[( 3x + 2) ^ 2 = 6x + 4\]
Now, expand and simplify the left-hand side:
\[( 3x + 2) ^ 2 = 9x^ 2 + 12x + 4\]
So, we have:
\[9x^ 2 + 12x + 4 = 6x + 4\]
Subtract \(6x + 4\) from both sides:
\[9x^ 2 + 12x + 4 - 6x - 4 = 0\]
Simplify:
\[9x^ 2 + 6x = 0\]
Factor out \(3x\):
\[3x( 3x + 2) = 0\]
So, the quadratic equation is:
\[3x( 3x + 2) = 0\]
This equation can be solved by setting each factor to zero:
\[3x = 0 \quad \text { or} \quad 3x + 2 = 0\]
Solving these, we get:
\[x = 0 \quad \text { or} \quad x = - \frac { 2} { 3} \]
Thus, the quadratic equation formed is:
\[3x( 3x + 2) = 0 \quad \text { or} \quad 9x^ 2 + 6x = 0.\]
 

Supplemental Knowledge

To solve this problem, we need to understand the area calculations for both a square and a right-angled triangle.

Area of a Square: The area of a square is given by the formula \(\text { side} ^ 2\). If the side length of the square is \(s\), then the area \(A\) is:
\[A = s^ 2\]

Area of a Right-Angled Triangle: The area of a right-angled triangle is given by \(\frac { 1} { 2} \times \text { base} \times \text { height} \). If the base is \(b\) and the height is \(h\), then the area \(A' \) is:
\[A' = \frac { 1} { 2} bh\]
Given that both areas are equal, we can set up an equation to solve for \(x\).

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