Solve for the roots in simplest form using the quadratic formula:
3, x, squared, minus, 36, x, equals, minus, 93
3x 
2
 −36x=
−93

To solve the quadratic equation using the quadratic formula, we first need to rewrite the equation in standard form:

\[
3x^2 - 36x + 93 = 0
\]

The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(3x^2 - 36x + 93 = 0\):

- \(a = 3\)
- \(b = -36\)
- \(c = 93\)

Now, plug these values into the quadratic formula:

\[
x = \frac{-(-36) \pm \sqrt{(-36)^2 - 4(3)(93)}}{2(3)}
\]

Simplifying step by step:

1. **Calculate the discriminant** (\(\Delta\)):
   \[
   \Delta = (-36)^2 - 4(3)(93) = 1296 - 1116 = 180
   \]

2. **Substitute the discriminant** into the quadratic formula:
   \[
   x = \frac{36 \pm \sqrt{180}}{6}
   \]

3. **Simplify the square root** of the discriminant:
   \[
   \sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}
   \]

   So the formula becomes:
   \[
   x = \frac{36 \pm 6\sqrt{5}}{6}
   \]

4. **Simplify the expression**:
   \[
   x = 6 \pm \sqrt{5}
   \]

Therefore, the roots of the equation in simplest form are:

\[
x = 6 + \sqrt{5} \quad \text{or} \quad x = 6 - \sqrt{5}
\]

Would you like to explore the discriminant more deeply or have any questions?

Here are 5 questions you might find interesting:

1. How would the roots change if the coefficient of \(x^2\) were different?
2. What happens to the discriminant if the constant term \(c\) is altered?
3. Can you find the vertex of the parabola given by the equation \(3x^2 - 36x + 93 = 0\)?
4. How would the quadratic formula look if we had complex roots?
5. How can we verify the roots by substituting them back into the original equation?

**Tip:** The discriminant (\(b^2 - 4ac\)) can tell you the nature of the roots: if it's positive, there are two real roots; if zero, there's one real root; and if negative, the roots are complex.

Solve for the roots in simplest form using the quadratic formula: 3x^2 - 36x = -93

Learn how to solve the quadratic equation 3x^2 - 36x + 93 = 0 using the quadratic formula. This guide provides a step-by-step solution, discussing the calculation of the discriminant and simplification of roots. Ideal for high school students in Grades 9-12.

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