given the roots below, write the quadratic equation form ax²+bx+c=0 a.5 and 9 b.-⅔ and ¾ c.2+-√3 over 3

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.

Learn how to write quadratic equations in the form ax² + bx + c = 0 from given roots: 5 and 9, -⅔ and ¾, and 2 ± √3 / 3. Detailed step-by-step solutions provided for each case.

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