solve : X²-2ax + a² -1 = 0
Answers
Required Answer:-
Given:
- x² - 2ax + a² - 1 = 0
To Find:
- The values of x.
Solution:
Given,
➡ x² - 2ax + (a² - 1) = 0
➡ x² - (2a)x + (a + 1)(a - 1) = 0
➡ x² - (a + a)x + (a + 1)(a - 1) = 0
➡ x² - (a + 1 + a - 1)x + (a + 1)(a - 1) = 0
➡ x² - (a + 1)x - (a - 1)x + (a + 1)(a - 1) = 0
➡ x[x - (a + 1)] - (a - 1)[x - (a + 1)] = 0
➡ [x - (a - 1)][x - (a + 1)] = 0
➡ (x - a + 1)(x - a - 1) = 0
By zero product rule,
➡ Either (x - a + 1) = 0 or (x - a - 1) = 0
➡ x = a - 1, a + 1
★ Hence, the values of x are (a - 1) and (a + 1)
Alternative method,
Given,
➡ x² - 2ax + a² - 1 = 0
➡ x² - 2ax + a² = 1
➡ (x)² - 2 × (a) × (x) + (a)² = 1
➡ (x - a)² = 1 [(a - b)² = a² - 2ab + b²]
➡ (x - a) = √1
There are two values of √1, i.e, -1 and 1.
(-1)² = 1 and (1)² = 1
So, when (x - a) = 1
➡ x = a + 1
When (x - a) = -1
➡ x = a - 1
★ Hence, the values of x are (a + 1) and (a - 1).
Answer:
- x = a + 1, a - 1