Solve the equation $$x^2 - 11x + 18 = 0$$. If the equation has more than one root, write the larger root in the answer.

Explanation:

We need to solve the quadratic equation $$x^2 - 11x + 18 = 0$$. We can use the quadratic formula or factoring to find the roots. Since we need to provide the larger root if there are multiple, factoring is often quicker if it's straightforward.

Step-by-step solution:

1. The equation is in the standard quadratic form $$ax^2 + bx + c = 0$$, where $$a=1$$, $$b=-11$$, and $$c=18$$.
2. We can try to factor the quadratic expression. We are looking for two numbers that multiply to $$c$$ (18) and add up to $$b$$ (-11).
3. Let's list the pairs of factors of 18: (1, 18), (2, 9), (3, 6).
4. Now let's consider the sums of these pairs. To get a sum of -11, both numbers must be negative.
5. The pairs of negative factors are: (-1, -18), (-2, -9), (-3, -6).
6. Let's check their sums:
• -1 + (-18) = -19
• -2 + (-9) = -11
• -3 + (-6) = -9
7. The pair (-2, -9) satisfies both conditions: $$(-2) \times (-9) = 18$$ and $$(-2) + (-9) = -11$$.
8. So, we can factor the quadratic equation as $$(x - 2)(x - 9) = 0$$.
9. To find the roots, we set each factor equal to zero:
• $$x - 2 = 0 \rightarrow x = 2$$
• $$x - 9 = 0 \rightarrow x = 9$$
10. The roots of the equation are 2 and 9.
11. The problem asks for the larger root. Comparing 2 and 9, the larger root is 9.

Answer: 9