Let's analyze each problem step by step.

16. Quadratic equation with roots -3 and -4

To form a quadratic equation from the roots, you use the fact that if $r_1$ and $r_2$ are the roots, the equation can be written as:

$(x - r_1)(x - r_2) = 0$

Here, $r_1 = -3$ and $r_2 = -4$. Substituting these into the equation:

$(x + 3)(x + 4) = 0$

Expanding this:

$x^2 + 4x + 3x + 12 = 0$

Simplifying:

$x^2 + 7x + 12 = 0$

Correct answer: $\mathbf{x^2 + 7x + 12 = 0}$

17. Equation representing the garden dimensions

Let the width be $x$, and the length is 7 meters longer, so the length is $x + 7$. The area of the garden is given as 18 square meters. The area of a rectangle is given by:

$\text{Area} = \text{Length} \times \text{Width}$

Substituting the values:

$x(x + 7) = 18$

Correct answer: $\mathbf{x(x + 7) = 18}$

18. Finding the length of the garden

Using the equation $x(x + 7) = 18$, we solve for $x$ (the width).

$x^2 + 7x - 18 = 0$

This quadratic equation can be solved by factoring or using the quadratic formula:

$x = \frac{-7 \pm \sqrt{7^2 + 4 \times 18}}{2 \times 1}$

The width $x = 2$ meters, so the length is:

$x + 7 = 2 + 7 = 9 \text{ meters}$

Correct answer: $\mathbf{9 \text{ meters}}$

19. Equation for the playground dimensions

Let the width be $w$, and the length is 25 meters longer, so the length is $w + 25$. The area of the playground is 3000 square meters. The equation becomes:

$w(w + 25) = 3000$

Correct answer: $\mathbf{w(w + 25) = 3000}$

20. Finding the dimensions of the playground

Using the equation $w(w + 25) = 3000$, solve for $w$:

$w^2 + 25w - 3000 = 0$

Factoring or using the quadratic formula:

$w = \frac{-25 \pm \sqrt{25^2 + 4 \times 3000}}{2 \times 1}$

The correct solution gives $w = 40$ meters. The length is $w + 25 = 65$ meters.

Correct answer: $\mathbf{40 \text{ meters by } 75 \text{ meters}}$

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Would you like more detailed steps or have any other questions? Here are some related questions to consider:

1. How do you derive the quadratic formula?
2. What are some methods for solving quadratic equations?
3. Can the dimensions of a rectangle always be determined from the area alone?
4. How can you confirm that a given quadratic equation has real roots?
5. What are the key properties of a quadratic equation?

Tip: For quadratic equations, always check if they can be factored easily before using the quadratic formula.