To factor the quadratic equation \(x^2 + 6x + 7 = 0\), we first try to express it in the form:

\[
(x + p)(x + q) = 0
\]

where \(p\) and \(q\) are numbers that satisfy:

\[
p + q = 6 \quad \text{and} \quad pq = 7
\]

However, after checking the possible factors, we find that no such integers \(p\) and \(q\) exist. Therefore, this quadratic expression is not factorable over the integers. 

To solve it, we can either use the quadratic formula:

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

For the given equation \(x^2 + 6x + 7 = 0\), \(a = 1\), \(b = 6\), and \(c = 7\). Plugging in these values:

\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(7)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 - 28}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{8}}{2}
\]
\[
x = \frac{-6 \pm 2\sqrt{2}}{2}
\]
\[
x = -3 \pm \sqrt{2}
\]

So, the roots of the equation are:

\[
x = -3 + \sqrt{2} \quad \text{or} \quad x = -3 - \sqrt{2}
\]

Thus, the factorized form over the real numbers is:

\[
(x + 3 - \sqrt{2})(x + 3 + \sqrt{2}) = 0
\]

Would you like more details on any part of this process or have any questions?

---

Here are five related questions:
1. How do you factorize a quadratic expression when it has rational roots?
2. What is the significance of the discriminant in solving quadratic equations?
3. Can all quadratic equations be factored? If not, why?
4. How would you solve a quadratic equation that cannot be factored?
5. What are some real-world applications of solving quadratic equations?

**Tip:** If a quadratic equation cannot be factored easily, the quadratic formula is a reliable alternative method for finding its roots.

Learn how to factorize and solve the quadratic equation x^2 + 6x + 7 = 0. Understand the process of factoring quadratic expressions and applying the quadratic formula to find the roots. Suitable for high school students studying quadratic equations and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation