If a - 1/a = 2 then show that
( a + 1/a )^2 = 8
To prove that (a + 1/a)^2 = 8, we can start by expanding the squared expression. Let's break down the steps to find the proof:
Step 1: Start with the given equation: a - 1/a = 2.
Step 2: To eliminate the fraction in the equation, we can multiply the entire equation by 'a' (since a * 1/a = 1). This gives us a^2 - 1 = 2a.
Step 3: Rearrange the equation to make it easier to work with: a^2 - 2a - 1 = 0.
Step 4: Now, let's find the values of 'a' using the quadratic formula: a = (-b ± sqrt(b^2 - 4ac)) / 2a, where a = 1, b = -2, and c = -1.
Step 5: Plugging in the values into the quadratic formula, we get a = (2 ± sqrt(4 + 4))/2, which simplifies to a = (2 ± sqrt(8))/2.
Step 6: Simplify the expression further: a = 1 ± √2.
Step 7: We have found two possible values for 'a': a = 1 + √2 and a = 1 - √2.
Now, let's substitute these two possible values of 'a' into the expression (a + 1/a)^2 and check if it equals 8.
Case 1: a = 1 + √2.
Substituting it into the expression: (1 + √2 + 1/(1 + √2))^2.
Step 8: To make the expression easier to work with, let's rationalize the denominator by multiplying both the numerator and denominator of the fraction by (1 - √2).
So, the expression becomes ((1 + √2)(1 - √2) + 1) / (1 + √2))^2.
Simplifying further, we get ((1 - 2) + 1) / (1 + √2))^2 = 0 / (1 + √2))^2 = 0.
Case 2: a = 1 - √2.
Substituting it into the expression: (1 - √2 + 1/(1 - √2))^2.
Step 9: Rationalize the denominator: ((1 - √2)(1 + √2) + 1) / (1 - √2))^2.
Simplifying further, we get ((1 - 2) + 1) / (1 - √2))^2 = 0 / (1 - √2))^2 = 0.
After evaluating both cases, we find that regardless of the values of 'a', the expression (a + 1/a)^2 is always equal to 0, not 8. Therefore, the statement (a + 1/a)^2 = 8 is not valid, given the equation a - 1/a = 2.