solve the quadratic equation 3x2 + x - 5 = 0 using the quadratic formula

To solve the quadratic equation \( 3x^2 + x - 5 = 0 \) using the quadratic formula, follow these steps:

The quadratic formula is given by:

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

Where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). 

In the equation \( 3x^2 + x - 5 = 0 \), the coefficients are:
- \( a = 3 \)
- \( b = 1 \)
- \( c = -5 \)

Now, substitute these values into the quadratic formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(3)(-5)}}{2(3)}
\]

Step-by-step solution:

1. Calculate the discriminant:
   \[
   \Delta = b^2 - 4ac = 1^2 - 4(3)(-5) = 1 + 60 = 61
   \]

2. Now substitute the discriminant back into the formula:
   \[
   x = \frac{-1 \pm \sqrt{61}}{6}
   \]

3. Simplify the expression:
   \[
   x_1 = \frac{-1 + \sqrt{61}}{6}, \quad x_2 = \frac{-1 - \sqrt{61}}{6}
   \]

Therefore, the two solutions are:
\[
x_1 = \frac{-1 + \sqrt{61}}{6}, \quad x_2 = \frac{-1 - \sqrt{61}}{6}
\]

These are the exact solutions. If you need approximate decimal values:
- \( x_1 \approx 1.280 \)
- \( x_2 \approx -1.947 \)

Would you like any further details or clarifications?

Here are 5 related questions to expand on this:
1. How do you determine the nature of the roots using the discriminant?
2. What happens when the discriminant is zero or negative?
3. Can you solve the quadratic equation using factoring instead of the quadratic formula?
4. How do you solve quadratic equations by completing the square?
5. What is the significance of the vertex in a quadratic function?

**Tip:** Always check the discriminant (\( \Delta = b^2 - 4ac \)) first to determine if the solutions are real, imaginary, or rational.

Discover how to solve the quadratic equation 3x^2 + x - 5 using the quadratic formula with a step-by-step solution. This explanation covers key algebraic concepts, including calculating the discriminant and using the quadratic formula to find the roots. Suitable for students in Grades 8-10.

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