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solve each problem with the quadratic formula

The equation provided is:

\[
x^2 + 7x - 4 = 0
\]

To solve this quadratic equation using the quadratic formula, we use the formula:

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

Where the coefficients from the equation \(ax^2 + bx + c = 0\) are:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 7\) (coefficient of \(x\))
- \(c = -4\) (constant term)

### Step 1: Calculate the discriminant
\[
\Delta = b^2 - 4ac = 7^2 - 4(1)(-4) = 49 + 16 = 65
\]

### Step 2: Apply the quadratic formula
\[
x = \frac{-7 \pm \sqrt{65}}{2(1)} = \frac{-7 \pm \sqrt{65}}{2}
\]

Thus, the solutions are:
\[
x_1 = \frac{-7 + \sqrt{65}}{2}
\]
\[
x_2 = \frac{-7 - \sqrt{65}}{2}
\]

These are the exact solutions. To get numerical approximations:

\[
\sqrt{65} \approx 8.062
\]
\[
x_1 \approx \frac{-7 + 8.062}{2} \approx \frac{1.062}{2} \approx 0.531
\]
\[
x_2 \approx \frac{-7 - 8.062}{2} \approx \frac{-15.062}{2} \approx -7.531
\]

### Final solutions:
\[
x_1 \approx 0.531
\]
\[
x_2 \approx -7.531
\]

Would you like more details on any step or further clarification?

Here are 5 related questions:
1. What is the discriminant and how does it determine the nature of the solutions?
2. How does the quadratic formula work for equations with complex solutions?
3. What methods other than the quadratic formula can solve quadratic equations?
4. How do you graph the solutions of a quadratic equation on the coordinate plane?
5. What happens to the solutions if the coefficient \(a\) is not 1?

**Tip:** Always check the discriminant before solving using the quadratic formula to determine if you should expect real or complex solutions.

Solve the quadratic equation: x^2 + 7x - 4 = 0

Step-by-step guide on how to solve the quadratic equation x^2 + 7x - 4 using the quadratic formula. This explanation includes calculating the discriminant and finding both the exact and approximate solutions. Ideal for students in Grades 8-10.

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