Question

The number of real solutions of the equation \((x^2 - 15x + 55)^{x^2 - 5x + 6} = 1\) is __________.

Hint:

When dealing with exponential equations, consider all possible cases for the base and exponent to find all real solutions. Ensure that the base is non-zero when the exponent is zero.

Answer 1

Step 1: Understand the general form of the equation.

The equation (x² - 15x + 55)^{x2 - 5x + 6} = 1 will hold in the following cases:

• If the base is 1, i.e., x² - 15x + 55 = 1.
• If the base is -1, i.e., x² - 15x + 55 = -1, and the exponent is an even number.
• If the exponent is 0, i.e., x² - 5x + 6 = 0, and the base is non-zero.

Step 2: Case 1 - Base is 1.

First, solve for x when x² - 15x + 55 = 1:

x² - 15x + 55 = 1 → x² - 15x + 54 = 0

Solving this quadratic equation:

x = (-(-15) ± √((-15)² - 4(1)(54))) / 2(1) = (15 ± √(225 - 216)) / 2 = (15 ± √9) / 2 = (15 ± 3) / 2

Thus, the solutions are:

x = (15 + 3) / 2 = 9 or x = (15 - 3) / 2 = 6

So, x = 9 and x = 6 are solutions.

Step 3: Case 2 - Base is -1 and the exponent is even.

Now solve for x when x² - 15x + 55 = -1:

x² - 15x + 55 = -1 → x² - 15x + 56 = 0

Solving this quadratic equation:

x = (-(-15) ± √((-15)² - 4(1)(56))) / 2(1) = (15 ± √(225 - 224)) / 2 = (15 ± √1) / 2 = (15 ± 1) / 2

Thus, the solutions are:

x = (15 + 1) / 2 = 8 or x = (15 - 1) / 2 = 7

For both of these values of x, check if x² - 5x + 6 is even:

For x = 8:

x² - 5x + 6 = 8² - 5(8) + 6 = 64 - 40 + 6 = 30 (even)

For x = 7:

x² - 5x + 6 = 7² - 5(7) + 6 = 49 - 35 + 6 = 20 (even)

So, x = 7 and x = 8 are solutions.

Step 4: Case 3 - Exponent is 0.

Solve for x when x² - 5x + 6 = 0:

x² - 5x + 6 = 0

Factoring the quadratic:

(x - 2)(x - 3) = 0

Thus, the solutions are:

x = 2 or x = 3

For both of these values of x, check that the base x² - 15x + 55 is non-zero:

For x = 2:

x² - 15x + 55 = 2² - 15(2) + 55 = 4 - 30 + 55 = 29 (non-zero)

For x = 3:

x² - 15x + 55 = 3² - 15(3) + 55 = 9 - 45 + 55 = 19 (non-zero)

So, x = 2 and x = 3 are solutions.

Step 5: Conclusion.

Thus, the real solutions are:

x = 9, 6, 8, 7, 2, 3

The total number of real solutions is:

6