To solve the quadratic equation $x^2 + 6x + 7 = 0$, we can use the quadratic formula:

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

In this case, the equation is of the form $ax^2 + bx + c = 0$, where:

• $a = 1$
• $b = 6$
• $c = 7$

Step 1: Calculate the discriminant $\Delta$

The discriminant is given by:

$\Delta = b^2 - 4ac$

Substitute the values:

$\Delta = 6^2 - 4(1)(7) = 36 - 28 = 8$

Step 2: Apply the quadratic formula

Since the discriminant is positive, the equation has two distinct real roots:

$x = \frac{-6 \pm \sqrt{8}}{2(1)}$

Simplify $\sqrt{8} = 2\sqrt{2}$:

$x = \frac{-6 \pm 2\sqrt{2}}{2}$

Now simplify further:

$x = \frac{-6}{2} \pm \frac{2\sqrt{2}}{2}$

$x = -3 \pm \sqrt{2}$

Final Answer:

The two solutions are:

$x_1 = -3 + \sqrt{2}, \quad x_2 = -3 - \sqrt{2}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you determine the number of solutions in a quadratic equation?
2. What happens if the discriminant is zero?
3. Can a quadratic equation have complex roots? When?
4. How can you graph a quadratic equation and what does it represent?
5. How do you factor quadratic equations with integer solutions?

Tip: Always check the discriminant first—it tells you if the roots are real or complex.