Explanation

Given polynomial: p(x) = x² - 4x + 3

Step 1: Factor the polynomial

p(x) = (x - 3)(x - 1)

Step 2: Find the zeros

Zeros are x = 3 and x = 1

Explanation

Given quadratic equation: 9x² - 27x + k = 0

Step 1: Find the sum and product of roots

Sum of roots (α + β) = -(-27)/9 = 3

Product of roots (αβ) = k/9

Step 2: Use the given condition

2α + 5β = 7

We also know α + β = 3

Step 3: Express α in terms of β or vice versa

From α + β = 3, α = 3 - β

Step 4: Substitute α in the given condition

2(3 - β) + 5β = 7

6 - 2β + 5β = 7

3β = 1

β = 1/3

Step 5: Find α

α = 3 - β = 3 - 1/3 = 8/3

Step 6: Find k using the product of roots

αβ = k/9

(8/3)(1/3) = k/9

8/9 = k/9

k = 8

Explanation

Z1Z2 = (2 - i)(3 - i)

= 6 - 2i - 3i + i²

= 6 - 5i - 1 (since i² = -1)

= 5 - 5i

Explanation

Given log₁₂(12³) = x:

Using logarithm properties, logₐ(aⁿ) = n:

x = 3

Explanation

Given b^n = b^m, for b ≠ 0:

Step 1: Analyze the equation

If b^n = b^m, then n = m, or b = 1 or b = -1 with specific conditions on n and m.

Step 2: Consider the case when b ≠ 1 and b ≠ -1

For b ≠ 0, 1, or -1, the equation holds true when n = m.

Step 3: Look for an option that satisfies n = m

Among the given options, one possible solution where n = m is when both n and m are 0, since any non-zero number to the power of 0 is 1.

The answer is: m = 0 and n = 0.