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The given equation is:

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

To solve this using the quadratic formula, let's first rearrange it into the standard quadratic form:

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

The quadratic formula is given by:

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

For the equation \(x^2 + 7x - 4 = 0\), we can identify the coefficients:
- \(a = 1\)
- \(b = 7\)
- \(c = -4\)

Now, we substitute these values into the quadratic formula:

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(1)(-4)}}{2(1)}
\]

Simplifying the discriminant:

\[
x = \frac{-7 \pm \sqrt{49 + 16}}{2}
\]
\[
x = \frac{-7 \pm \sqrt{65}}{2}
\]

Thus, the solutions are:

\[
x = \frac{-7 + \sqrt{65}}{2} \quad \text{or} \quad x = \frac{-7 - \sqrt{65}}{2}
\]

These are the two roots of the quadratic equation.

Would you like more details on any of the steps, or clarification on the quadratic formula? 

Here are some related questions:
1. How do you determine when to use the quadratic formula?
2. What is the significance of the discriminant in quadratic equations?
3. Can a quadratic equation have complex solutions, and how does that occur?
4. How can completing the square be used to solve this quadratic equation?
5. What are real-life applications of quadratic equations?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before solving; it tells you if the roots will be real or complex.

Solve by using the quadratic formula for the equation: x^2 + 7x = 4

Solve the quadratic equation x^2 + 7x - 4 = 0 using the quadratic formula, with a detailed breakdown of the steps involved. Learn how to apply the formula to find the roots, and understand key concepts like the discriminant. Suitable for students in Grades 9-11.

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