Explain how to solve the quadratic equation x^2 - 4x - 5 = 0 using the math solver with steps.

To solve the quadratic equation \( x^2 - 4x - 5 = 0 \) using a step-by-step approach, you can follow the steps below:

### Step 1: Identify the coefficients
For a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -4 \) (coefficient of \( x \))
- \( c = -5 \) (constant term)

### Step 2: Calculate the discriminant
The discriminant \( \Delta \) is given by the formula:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
\Delta = (-4)^2 - 4(1)(-5) = 16 + 20 = 36
\]

### Step 3: Determine the nature of the roots
The discriminant \( \Delta = 36 \) is positive, indicating that the quadratic equation has two distinct real roots.

### Step 4: Apply the quadratic formula
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\]
Substitute the values of \( a \), \( b \), and \( \Delta \):
\[
x = \frac{-(-4) \pm \sqrt{36}}{2(1)} = \frac{4 \pm 6}{2}
\]

### Step 5: Solve for the roots
Now, calculate the two possible values for \( x \):

1. **First root:**
   \[
   x_1 = \frac{4 + 6}{2} = \frac{10}{2} = 5
   \]

2. **Second root:**
   \[
   x_2 = \frac{4 - 6}{2} = \frac{-2}{2} = -1
   \]

### Step 6: State the solution
The solutions to the quadratic equation \( x^2 - 4x - 5 = 0 \) are:
\[
x = 5 \quad \text{and} \quad x = -1
\]

These are the values of \( x \) that satisfy the equation.

Would you like more details or have any questions about this method? 

Here are 8 related questions to explore:

1. How do you complete the square to solve a quadratic equation?
2. What is the significance of the discriminant in a quadratic equation?
3. Can the quadratic formula be used for equations where \( a \neq 1 \)?
4. How do you factorize the quadratic equation \( x^2 - 4x - 5 \)?
5. What happens if the discriminant is zero?
6. How do you graph the quadratic function \( y = x^2 - 4x - 5 \)?
7. How can you derive the quadratic formula from the general quadratic equation?
8. What are the applications of quadratic equations in real life?

**Tip:** Always check the discriminant first; it gives you insight into the nature of the roots before you solve the equation.

Learn how to solve the quadratic equation x^2 - 4x - 5 = 0 using a step-by-step method. Understand the calculation of discriminant and application of the quadratic formula. Suitable for high school students (Grades 10-12) studying algebra and quadratic equations.

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