Explanation:

Let’s break this down step by step:

- First, we solve for x:

The equation is x² – 10x + 21 = 0.

This factors to (x – 7)(x – 3) = 0, so x = 3 or 7.

Take the smaller root, so x = 3.

- Now for y:

y² – 33y + 266 = 0.

Factors: (y – 14)(y – 19) = 0, so y = 14 or 19.

Take the greater root, so y = 19.

- Next, z:

z = 9x – 1 = 9*3 – 1 = 26.

Now let’s plug these values into the formulas for A, B, C, D:

- A: (x + 1)² + y = (3 + 1)² + 19 = 16 + 19 = 35

- B: (x + 2)² + z = (3 + 2)² + 26 = 25 + 26 = 51

- C: (x + 6)² + 2 = (3 + 6)² + 2 = 81 + 2 = 83

- D: (x + 9)² + 3 = (3 + 9)² + 3 = 144 + 3 = 147

So, the series is: ?, 35, 51, 83, 147.

Looks like there’s a pattern, and we need to figure out what comes before 35.

Let’s see the difference pattern:

- 35 – ? = ???

- 51 – 35 = 16

- 83 – 51 = 32

- 147 – 83 = 64

The differences are 16, 32, 64 — a doubling pattern, so before 16 should be 8.

So, 35 – 8 = 27.

Now the options:

- 1. 19

- 2. 27

- 3. 29

- 4. 35

- 5. 17

The correct answer is option 2: 27.

27

To sum up:

- You use the roots for x and y as the question says.

- You substitute everything step by step.

- The pattern in the sequence differences shows where 27 fits.

- Don’t overthink it—the doubling difference gives it away.