To solve the quadratic equation \( x^2 - 14x + 2 = 0 \), we will use the quadratic formula:

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

For the given equation, we have:
- \( a = 1 \) (coefficient of \( x^2 \)),
- \( b = -14 \) (coefficient of \( x \)),
- \( c = 2 \) (constant term).

Now, substitute these values into the quadratic formula:

\[
x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(2)}}{2(1)}
\]
\[
x = \frac{14 \pm \sqrt{196 - 8}}{2}
\]
\[
x = \frac{14 \pm \sqrt{188}}{2}
\]
\[
x = \frac{14 \pm \sqrt{4 \times 47}}{2}
\]
\[
x = \frac{14 \pm 2\sqrt{47}}{2}
\]
\[
x = 7 \pm \sqrt{47}
\]

Thus, the two solutions are:

\[
x = 7 + \sqrt{47} \quad \text{or} \quad x = 7 - \sqrt{47}
\]

Would you like to see the decimal approximation for these solutions or a more detailed breakdown of any step?

Here are 5 related questions:
1. How does the discriminant affect the nature of the roots?
2. What is the quadratic formula, and when should it be used?
3. How do you factor quadratics when the discriminant is a perfect square?
4. Can all quadratic equations be solved using factoring?
5. What is the significance of the vertex form of a quadratic equation?

**Tip:** Always check the discriminant \( b^2 - 4ac \) first to determine if the quadratic has real, rational, or complex roots.

Learn how to solve the quadratic equation x^2 - 14x + 2 using the quadratic formula. This detailed step-by-step solution includes key algebraic concepts such as the quadratic formula and the properties of the discriminant. Suitable for students in Grades 9-12.

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