sovle for the quadratic formula, 4x+19x-5=0

To solve the quadratic equation \( 4x^2 + 19x - 5 = 0 \), we will use the **quadratic formula**, which is given by:

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

For the equation \( 4x^2 + 19x - 5 = 0 \), we can identify the coefficients as:
- \( a = 4 \)
- \( b = 19 \)
- \( c = -5 \)

### Step 1: Calculate the discriminant
The discriminant \( \Delta \) is calculated as:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
\Delta = 19^2 - 4(4)(-5) = 361 + 80 = 441
\]

### Step 2: Apply the quadratic formula
Now, we substitute the values into the quadratic formula:
\[
x = \frac{-19 \pm \sqrt{441}}{2(4)} = \frac{-19 \pm 21}{8}
\]

### Step 3: Solve for the two possible solutions
- \( x_1 = \frac{-19 + 21}{8} = \frac{2}{8} = \frac{1}{4} \)
- \( x_2 = \frac{-19 - 21}{8} = \frac{-40}{8} = -5 \)

### Final Answer:
The solutions are:
\[
x_1 = \frac{1}{4}, \quad x_2 = -5
\]

Would you like further details or explanations on any of the steps?

Here are 5 related questions to consider:
1. How do you find the discriminant of a quadratic equation?
2. What does the discriminant tell you about the nature of the solutions?
3. Can a quadratic equation have complex solutions?
4. How do you graph a quadratic equation and find its vertex?
5. How does completing the square compare with the quadratic formula?

**Tip:** Always check the discriminant before solving; it helps you predict the type of solutions (real or complex).

Discover how to solve the quadratic equation 4x² + 19x - 5 = 0 using the quadratic formula. This detailed explanation includes calculating the discriminant, applying the formula, and finding the roots. Ideal for students in Grades 9-11.

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