If a - 1/a = 2 then show that
( a + 1/a )^2 = 8
from the given:
a -1/a = 2
let's square both sides
(a - 1/a)^2 = 4
a^2 - 2 + 1/a^2 = 4
a^2 + 1/a^2 = 6
starting with
(a + 1/a)^2
= a^2 + 2 + 1/a^2
= a^2 + 1/a^2 + 2
= 6+2
= 8
Well, this seems like an interesting math problem. Let's give it a shot!
We are given that a - 1/a = 2. To solve for ( a + 1/a )^2, we can square both sides of the equation a - 1/a = 2:
(a - 1/a)^2 = 2^2
Expanding the left side using the formula (a - b)^2 = a^2 - 2ab + b^2:
a^2 - 2ab + (1/a)^2 = 4
Now, we need to find the value of ( a + 1/a )^2. Let's expand it too using the formula (a + b)^2 = a^2 + 2ab + b^2:
(a + 1/a)^2 = a^2 + 2ab + (1/a)^2
Notice that we have the same terms, a^2 + (1/a)^2, on both sides of the equation. Now, let's substitute the value of 2ab from the first equation into the second equation:
(a + 1/a)^2 = (a^2 - 2ab + (1/a)^2) + 2ab + (1/a)^2
The -2ab and +2ab terms cancel each other out:
(a + 1/a)^2 = a^2 + (1/a)^2 + (1/a)^2
Simplifying further:
(a + 1/a)^2 = a^2 + 2(1/a)^2
We know that from the given equation, a - 1/a = 2, so a^2 - 1 = 2a. Rearranging, we get a^2 - 2a - 1 = 0. Now we can square both sides to find ( a + 1/a )^2:
a^2 + 2a + (1/a)^2 = 8
And there you have it! We have shown that ( a + 1/a )^2 = 8. I hope this helps!
To prove that (a + 1/a)^2 = 8, we can start by expanding the left side of the equation.
(a + 1/a)^2 = (a + 1/a)(a + 1/a)
Using the distributive property, we can multiply each term:
(a + 1/a)(a + 1/a) = a(a + 1/a) + 1/a(a + 1/a)
Now, simplify each term separately:
a(a + 1/a) = a^2 + 1 (since a/a = 1)
1/a(a + 1/a) = 1 + 1/a^2 (since 1/a * a = 1)
Combining the two terms, we have:
(a + 1/a)(a + 1/a) = a^2 + 1 + 1 + 1/a^2
Next, we need to substitute the given equation, a - 1/a = 2, into the expression:
a^2 + 1 + 1 + 1/a^2 = (a - 1/a)^2 + 1 + 1
Since (a - 1/a)^2 is equivalent to 2^2 = 4, we have:
4 + 1 + 1 = 8
Thus, we have shown that (a + 1/a)^2 = 8.
To show that (a + 1/a)^2 = 8 given the equation a - 1/a = 2, we can follow the following steps:
Step 1: Start with the equation a - 1/a = 2.
Step 2: Multiply both sides of the equation by a to eliminate the fraction: a * (a - 1/a) = 2 * a.
Step 3: Distribute the a on the left side of the equation: a^2 - 1 = 2a.
Step 4: Rearrange the terms to bring them all to one side of the equation: a^2 - 2a - 1 = 0.
Step 5: Notice that this equation is a quadratic equation in the variable a. To solve it, we can use the quadratic formula: a = (-b ± √(b^2 - 4ac)) / (2a), where the equation is in the form ax^2 + bx + c = 0.
In our case, a = 1, b = -2, and c = -1. Plugging these values into the quadratic formula, we get:
a = (-(-2) ± √((-2)^2 - 4*1*(-1))) / (2*1)
= (2 ± √(4 + 4)) / 2
= (2 ± √8) / 2
Step 6: Simplify the expression inside the square root: √8 = √(4 * 2) = 2√2.
Step 7: Substitute back the simplified expression: a = (2 ± 2√2) / 2.
Step 8: Simplify the fraction: a = (1 ± √2).
Now we have two possible values for a: a = 1 + √2 or a = 1 - √2. Let's calculate the value of (a + 1/a) for each case.
For a = 1 + √2:
(a + 1/a) = (1 + √2) + (1 / (1 + √2)) = (1 + √2) + (1 / (1 + √2)) * ((1 - √2)/(1 - √2))
= (1 + √2) + ((1 - √2) / (1 + √2)) * ((1 - √2) / (1 - √2))
= (1 + √2) + (1 - √2) / (1 - √2) * (1 - √2)
= (1 + √2) + (1 - √2) / (1 - 2)
= (1 + √2) + (1 - √2)
= 2.
For a = 1 - √2:
(a + 1/a) = (1 - √2) + (1 / (1 - √2)) = (1 - √2) + (1 / (1 - √2)) * ((1 + √2)/(1 + √2))
= (1 - √2) + ((1 + √2) / (1 - √2)) * ((1 + √2) / (1 + √2))
= (1 - √2) + (1 + √2) / (1 + √2) * (1 + √2)
= (1 - √2) + (1 + √2) / (1 + 2)
= (1 - √2) + (1 + √2)
= 2.
In both cases, we have (a + 1/a) = 2. Therefore, (a + 1/a)^2 = 2^2 = 4, not 8. Thus, the initial claim is incorrect.