sovle for the quadratic formula, 4x^2+19x-5=0

To solve the quadratic equation \( 4x^2 + 19x - 5 = 0 \), we can use the quadratic formula:

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

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

Now, substitute these values into the quadratic formula:

\[
x = \frac{-19 \pm \sqrt{19^2 - 4(4)(-5)}}{2(4)}
\]

### Step 1: Calculate the discriminant
\[
b^2 - 4ac = 19^2 - 4(4)(-5) = 361 + 80 = 441
\]

### Step 2: Substitute the discriminant into the formula
\[
x = \frac{-19 \pm \sqrt{441}}{8}
\]

### Step 3: Simplify the square root
\[
\sqrt{441} = 21
\]

### Step 4: Find 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 solutions:
\[
x = \frac{1}{4} \quad \text{or} \quad x = -5
\]

Would you like a detailed explanation of any specific part? 

Here are some related questions to explore:
1. What is the significance of the discriminant in the quadratic formula?
2. How do you derive the quadratic formula from a general quadratic equation?
3. Can the quadratic formula be used for complex solutions, and how?
4. What happens if the discriminant is zero or negative?
5. How does factoring compare to the quadratic formula for solving equations?

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

Solve for the quadratic formula, 4x^2 + 19x - 5 = 0

Learn how to solve the quadratic equation 4x^2 + 19x - 5 = 0 using the quadratic formula. This step-by-step guide covers key concepts such as calculating the discriminant and finding real solutions. Suitable for students in Grades 9-11.

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