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Let's factorize the given expressions.

### (i) \( x^2 + 6x - 55 \)

We need to find two numbers that multiply to \(-55\) and add up to \(6\).

Factors of \(-55\) are:
- \(-1 \times 55\)
- \(1 \times -55\)
- \(-5 \times 11\)
- \(5 \times -11\)

Among these, the pair that adds up to \(6\) is \(11\) and \(-5\).

So, we can factorize the expression as:
\[
x^2 + 6x - 55 = (x + 11)(x - 5)
\]

### (ii) \( x^2 + \sqrt{3}x - 6 \)

Similarly, we need to find two numbers that multiply to \(-6\) and add up to \(\sqrt{3}\).

Let the two numbers be \( a \) and \( b \), such that:
\[
ab = -6 \quad \text{and} \quad a + b = \sqrt{3}
\]

We can solve this by trial and error, but it becomes apparent that a direct factorization approach might not be straightforward due to the irrational number \(\sqrt{3}\). Instead, we can use the quadratic formula to find the roots:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = \sqrt{3} \), and \( c = -6 \).

First, calculate the discriminant:
\[
\Delta = (\sqrt{3})^2 - 4(1)(-6) = 3 + 24 = 27
\]

So, the roots are:
\[
x = \frac{-\sqrt{3} \pm \sqrt{27}}{2} = \frac{-\sqrt{3} \pm 3\sqrt{3}}{2}
\]

This gives the roots:
\[
x = \frac{-\sqrt{3} + 3\sqrt{3}}{2} = \sqrt{3}, \quad x = \frac{-\sqrt{3} - 3\sqrt{3}}{2} = -2\sqrt{3}
\]

Thus, the factorization is:
\[
x^2 + \sqrt{3}x - 6 = (x - \sqrt{3})(x + 2\sqrt{3})
\]

Would you like any more details or explanations?

---

**Here are 5 related questions:**

1. How do you factorize a quadratic expression with non-integer roots?
2. What is the significance of the discriminant in the quadratic formula?
3. How do you handle the factorization of polynomials with irrational coefficients?
4. Can all quadratic equations be factorized into linear factors with real coefficients?
5. What methods can be used if the quadratic equation does not factor nicely?

**Tip:** Always check the discriminant before attempting to factorize a quadratic expression, as it will indicate whether the roots are real, equal, or complex.

Factorise: (i) x^2 + 6x - 55, (ii) x^2 + √3x - 6

Learn how to factorize quadratic equations such as x^2 + 6x - 55 and x^2 + √3x - 6. This problem covers key algebraic concepts including factorization, the quadratic formula, and handling irrational roots. Suitable for students in Grades 9-12.

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