The Quadratic Formula: Using additional geometric shapes

Since in a rectangle each pair of opposite sides are equal to each other, let's call each pair of sides X and Y

Now let's set up a formula to calculate the perimeter of the rectangle:

$2x+2y=14$

Let's divide both sides by 2:

$x+y=7$

From this formula, we'll calculate X:

$x=7-y$

We know that the area of the rectangle equals length times width:

$x\times y=12$

We know that X equals 7 minus Y, let's substitute this in the formula:

$(7-y)\times y=12$

$7y-y^2=12$

$y^2-7y+12=0$

$(y-3)\times(y-4)=0$

From this we can claim that:

$y=3,y=4$

Let's go back to the formula we found earlier:

$x=7-y$

Let's substitute y equals 3 and we get:

$x=7-3=4$

Now let's substitute y equals 4 and we get:

$x=7-4=3$

Therefore, the lengths of the rectangle's sides are 4 and 3

To determine the values of $X$ for which the given angle in the parallelogram is acute, we will follow these steps:

• Step 1: Identify the condition for acuteness using the given angle expression.
• Step 2: Solve the inequality to ensure the angle remains acute.
• Step 3: Analyze for any potential solutions or contradictions.

Now, let's carry out each step:
Step 1: The problem gives us the expression $5x - 42$ as the measurement of a labelled angle in the parallelogram. To remain acute, angles must satisfy the inequalities:

• $5x - 42 < 90$

Step 2: Solve the inequality: $5x - 42 < 90$ Adding 42 on both sides, we have: $5x < 132$ Dividing both sides by 5, we find: $x < 26.4$

Step 3: Since this angle is part of a parallelogram, the opposite angles ($180^\circ -$ measured angle) and adjacent angles also adhere to specific conditions. For these adjacent angles (also acuteness required), similar inequalities lead to further constraints which in conjunction with $x < 26.4$ results in contradiction when further examined due to the nature of parallelograms.

Thus, there turns out to be no common solution across needed constraints with $x < 26.4$.

Ultimately, no $X$ satisfies these conditions and keeps all angles in a parallelogram acute, confirming no solution exists for such a configuration under stated conditions.

Therefore, the solution to the problem is No solution.