The given quadratic equation is:

$x^2 + 4x - 7 = 0$

We will solve it using the quadratic formula, which is:

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

For the equation $x^2 + 4x - 7 = 0$, we have the following coefficients:

• $a = 1$
• $b = 4$
• $c = -7$

Step 1: Calculate the discriminant

The discriminant is given by $b^2 - 4ac$.

$\text{Discriminant} = 4^2 - 4(1)(-7) = 16 + 28 = 44$

Step 2: Apply the quadratic formula

Now, substitute the values into the quadratic formula:

$x = \frac{-4 \pm \sqrt{44}}{2(1)} = \frac{-4 \pm \sqrt{44}}{2}$

Since $\sqrt{44} = 2\sqrt{11}$, we simplify further:

$x = \frac{-4 \pm 2\sqrt{11}}{2}$

$x = -2 \pm \sqrt{11}$

Final Solution:

The two solutions are:

$x_1 = -2 + \sqrt{11}$ $x_2 = -2 - \sqrt{11}$

These are the exact solutions to the quadratic equation.

Would you like further details or have any questions? Here are 5 related questions to consider:

1. How do we determine the nature of the roots from the discriminant?
2. What is the significance of the quadratic formula in solving equations?
3. How can you approximate the solutions of the quadratic equation numerically?
4. Can you factor this quadratic equation instead of using the formula?
5. What are some practical applications of solving quadratic equations?

Tip: Always check the discriminant before solving; it gives insight into the number and type of solutions (real or complex).