To write the quadratic equation in the form $ax^2 + bx + c = 0$ given the roots, you can use the fact that if $p$ and $q$ are the roots, the equation can be written as:

$a(x - p)(x - q) = 0$

Expanding this, we get:

$ax^2 - a(p + q)x + apq = 0$

So, let's compute the quadratic equations for the given roots:

a. Roots: $5$ and $9$

Using the formula:

$(x - 5)(x - 9) = 0$

Expanding:

$x^2 - 9x - 5x + 45 = x^2 - 14x + 45 = 0$

So, the quadratic equation is:

$x^2 - 14x + 45 = 0$

b. Roots: $-\frac{2}{3}$ and $\frac{3}{4}$

Using the formula:

$\left( x + \frac{2}{3} \right)\left( x - \frac{3}{4} \right) = 0$

Expanding:

$x^2 - \frac{3}{4}x + \frac{2}{3}x - \frac{2}{3} \times \frac{3}{4} = 0$

$x^2 + \left(-\frac{9}{12} + \frac{8}{12}\right)x - \frac{6}{12} = 0$

$x^2 - \frac{1}{12}x - \frac{1}{2} = 0$

To eliminate fractions, multiply by 12:

$12x^2 - x - 6 = 0$

So, the quadratic equation is:

$12x^2 - x - 6 = 0$

c. Roots: $\frac{2 + \sqrt{3}}{3}$ and $\frac{2 - \sqrt{3}}{3}$

Using the formula:

$\left( x - \frac{2 + \sqrt{3}}{3} \right)\left( x - \frac{2 - \sqrt{3}}{3} \right) = 0$

This is a difference of squares:

$\left(x - \frac{2}{3} \right)^2 - \left(\frac{\sqrt{3}}{3}\right)^2 = 0$

Expanding:

$\left( x - \frac{2}{3} \right)^2 - \frac{3}{9} = 0$

$x^2 - \frac{4}{3}x + \frac{4}{9} - \frac{1}{3} = 0$

Simplifying:

$x^2 - \frac{4}{3}x + \frac{1}{9} = 0$

To eliminate fractions, multiply by 9:

$9x^2 - 12x + 1 = 0$

So, the quadratic equation is:

$9x^2 - 12x + 1 = 0$

Would you like more details or have any questions?

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Related Questions:

1. How do you derive the quadratic formula from the general form of a quadratic equation?
2. What is the discriminant and how does it relate to the nature of the roots of a quadratic equation?
3. How can you factor a quadratic equation if the roots are complex numbers?
4. What is the significance of the sum and product of the roots in a quadratic equation?
5. How do you solve a quadratic equation using the method of completing the square?

Tip: Always check your final quadratic equation by expanding the factors to ensure that it matches the original roots.