To solve the quadratic equation \( 5x^2 - x + 4 = 0 \) using the quadratic formula, follow these steps:

### Quadratic Formula
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Identify the coefficients
For the equation \( 5x^2 - x + 4 = 0 \):
- \( a = 5 \)
- \( b = -1 \)
- \( c = 4 \)

### Step 2: Calculate the discriminant
The discriminant is the part under the square root in the quadratic formula, \( b^2 - 4ac \):
\[
\text{Discriminant} = (-1)^2 - 4(5)(4) = 1 - 80 = -79
\]

### Step 3: Analyze the discriminant
Since the discriminant is negative (\( -79 \)), the equation has no real solutions, but it has two complex solutions.

### Step 4: Calculate the complex solutions
\[
x = \frac{-(-1) \pm \sqrt{-79}}{2(5)} = \frac{1 \pm \sqrt{79}i}{10}
\]

### Final Answer
The solutions to the equation \( 5x^2 - x + 4 = 0 \) are:
\[
x = \frac{1 + \sqrt{79}i}{10} \quad \text{and} \quad x = \frac{1 - \sqrt{79}i}{10}
\]

These are complex numbers since the discriminant is negative.

Would you like any further details or have any questions?

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Here are 5 related questions you might find interesting:

1. What do the different values of the discriminant indicate about the nature of the roots?
2. How can you solve a quadratic equation using the method of completing the square?
3. What are the applications of quadratic equations in real-life scenarios?
4. How do complex roots appear on the complex plane?
5. How do you factor a quadratic equation with complex roots?

**Tip:** When working with the quadratic formula, always check the discriminant first to determine the nature of the solutions (real or complex).

Learn how to solve the quadratic equation 5x² - x + 4 = 0 using the quadratic formula with a step-by-step explanation. Understand the nature of complex solutions and the discriminant. Suitable for high school students studying algebra.

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