To factorize the expression \(x^2 + 7\sqrt{6}x + 60\), we'll need to find two numbers that multiply to 60 and add to \(7\sqrt{6}\).

However, upon closer inspection, this does not seem to be straightforwardly factorable by simple methods due to the irrational coefficient \(7\sqrt{6}\). Let's use the quadratic formula to find the roots, then express the factorization in terms of these roots.

The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = 7\sqrt{6}\), and \(c = 60\).

Let's calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (7\sqrt{6})^2 - 4(1)(60) \]
\[ b^2 - 4ac = 49 \cdot 6 - 240 \]
\[ b^2 - 4ac = 294 - 240 \]
\[ b^2 - 4ac = 54 \]

Now, we can find the roots:
\[ x = \frac{-7\sqrt{6} \pm \sqrt{54}}{2} \]
\[ x = \frac{-7\sqrt{6} \pm 3\sqrt{6}}{2} \]
\[ x = \frac{-7\sqrt{6} + 3\sqrt{6}}{2} \quad \text{or} \quad x = \frac{-7\sqrt{6} - 3\sqrt{6}}{2} \]
\[ x = \frac{-4\sqrt{6}}{2} \quad \text{or} \quad x = \frac{-10\sqrt{6}}{2} \]
\[ x = -2\sqrt{6} \quad \text{or} \quad x = -5\sqrt{6} \]

Thus, the roots of the quadratic are \(-2\sqrt{6}\) and \(-5\sqrt{6}\).

We can write the factorized form of the quadratic as:
\[ (x + 2\sqrt{6})(x + 5\sqrt{6}) \]

So, the factorized form of \(x^2 + 7\sqrt{6}x + 60\) is:
\[ (x + 2\sqrt{6})(x + 5\sqrt{6}) \]

Would you like more details or have any questions?

Here are some questions you might ask next:
1. Can you explain how to derive the quadratic formula?
2. What are some methods to factorize quadratics with irrational coefficients?
3. How do you verify the factorization of a quadratic expression?
4. Can you solve a different quadratic equation using the quadratic formula?
5. What are some applications of quadratic equations in real life?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before attempting to factorize a quadratic equation. It can tell you if the roots are real, complex, or irrational, which can simplify your approach.

Learn how to factorize the quadratic expression x^2 + 7√6x + 60 using the quadratic formula and step-by-step method. This problem involves an irrational coefficient and demonstrates the application of the quadratic formula. Suitable for high school students in Grades 10-12.

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