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Answers
Answer:
6
Step-by-step explanation:
Given Equation is (x² - 7x + 11)ˣ²⁻¹³ˣ₊⁴² = 1
On applying 'log' on both sides, we get
⇒ (x² - 13x + 42)log(x² - 7x + 11) = log(1) {∵log aⁿ = n log a}
⇒ (x² - 13x + 42)log(x² - 7x + 11) = 0
Case (i)
x² - 13x + 42 = 0
⇒ x² - 7x - 6x + 42 = 0
⇒ x(x - 7) - 6(x - 7) = 0
⇒ (x - 6)(x - 7) = 0
⇒ x = 6, 7
Case (ii):
log(x² - 7x + 11) = 0
⇒ x² - 7x + 11 = 1
⇒ x² - 7x + 10 = 0
⇒ x² - 5x - 2x + 10 = 0
⇒ x(x - 5) - 2(x - 5) = 0
⇒ (x - 2)(x - 5) = 0
⇒ x = 2, 5
Case(iii):
x² - 7x + 11 = -1 {True, when Base is -1 and exponent is (-1)^2k}
⇒ x² - 7x + 12 = 0
⇒ x² - 4x - 3x + 12 = 0
⇒ x(x - 4) - 3(x - 4) = 0
⇒ (x - 3)(x - 4) = 0
⇒ x = 3,4
Now,
Plug x = 3 and x = 4 in exponent in order to verify that it is even.
(a) When x = 3
x² - 13x + 42 = 2k {k ∈ Z}
⇒ (3)² - 13(3) + 42 = 2k
⇒ 12 = 2k
∴ Valid
(b) When x = 4:
x² - 13x + 42 = 2k
⇒ (4)² - 13(4) + 42 = 2k
⇒ 16 - 52 + 42 = 2k
⇒ 6 = 2k
∴ Valid.
Thus, total number of solutions are 6(2,3,4,5,6,7)
Hope it helps!
Anything power 0 is 1.
So if solve the quadratic equation we will get 2,3,4,56,7
