To solve for the dimensions of the rectangular painting, we can set up a system of equations using the area and perimeter formulas.

Let the length of the rectangle be $l$ and the width be $w$.

Step 1: Write the area equation

The area $A$ of a rectangle is given by:

$A = l \times w$

We are told that the area is 35 square feet:

l \times w = 35 \tag{1}

Step 2: Write the perimeter equation

The perimeter $P$ of a rectangle is given by:

$P = 2l + 2w$

We are told that the perimeter is 24 feet:

$2l + 2w = 24$

Simplify the perimeter equation:

l + w = 12 \tag{2}

Step 3: Solve the system of equations

We now have the system of two equations:

1. $l \times w = 35$
2. $l + w = 12$

From equation (2), solve for $l$:

$l = 12 - w$

Substitute this into equation (1):

$(12 - w) \times w = 35$

Expand the equation:

$12w - w^2 = 35$

Rearrange the terms:

$w^2 - 12w + 35 = 0$

Step 4: Solve the quadratic equation

Solve the quadratic equation $w^2 - 12w + 35 = 0$ using the quadratic formula:

$w = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(35)}}{2(1)}$

$w = \frac{12 \pm \sqrt{144 - 140}}{2}$

$w = \frac{12 \pm \sqrt{4}}{2}$

$w = \frac{12 \pm 2}{2}$

This gives two solutions:

$w = \frac{12 + 2}{2} = 7 \quad \text{or} \quad w = \frac{12 - 2}{2} = 5$

Step 5: Find the corresponding lengths

For $w = 7$, from equation (2), $l = 12 - 7 = 5$.

For $w = 5$, from equation (2), $l = 12 - 5 = 7$.

Thus, the dimensions of the rectangle are:

$\boxed{5 \text{ feet} \times 7 \text{ feet}}$

These are the possible dimensions of the painting.

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Would you like further details or clarifications on any of the steps? Here are some related questions:

1. How can we use the quadratic formula to solve other real-world problems?
2. What would happen if the perimeter was increased to 30 feet?
3. How can we solve similar problems involving the area and perimeter of triangles?
4. What are some applications of quadratic equations in real-life scenarios?
5. How do you derive the quadratic formula?

Tip: When solving problems involving area and perimeter, always make sure to use the correct formulas and keep track of your units!