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.