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secondary 3 | A Maths
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Date Posted: 3 years ago
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x = 3 + 2√2
x = 2 + 2√2 + 1
x = (√2)² + 2√2(1) + 1²
Notice that this is in the form a² + 2ab + b², and we know that equals (a + b)². So we can rewrite it.
x = (√2 + 1)²
√x = √(√2 + 1)² = √2 + 1
Then,
√x - 1/√x = √2 + 1 - 1/(√2 + 1)
= √2 + 1 - (√2 - 1) / [(√2 + 1)(√2 - 1)]
= √2 + 1 - (√2 - 1) / ((√2)² - 1²)
= √2 + 1 - (√2 - 1) / (2 - 1)
= √2 + 1 - (√2 - 1)
= √2 + 1 - √2 + 1
= 2
Alternatively,
√x - 1/√x
= (√x√x - 1) / √x
= (x - 1) / √x
= (3 + 2√2 - 1) / (√2 + 1)
= (2√2 + 2) / (√2 + 1)
= 2(√2 + 1) / (√2 + 1)
= 2
x = 2 + 2√2 + 1
x = (√2)² + 2√2(1) + 1²
Notice that this is in the form a² + 2ab + b², and we know that equals (a + b)². So we can rewrite it.
x = (√2 + 1)²
√x = √(√2 + 1)² = √2 + 1
Then,
√x - 1/√x = √2 + 1 - 1/(√2 + 1)
= √2 + 1 - (√2 - 1) / [(√2 + 1)(√2 - 1)]
= √2 + 1 - (√2 - 1) / ((√2)² - 1²)
= √2 + 1 - (√2 - 1) / (2 - 1)
= √2 + 1 - (√2 - 1)
= √2 + 1 - √2 + 1
= 2
Alternatively,
√x - 1/√x
= (√x√x - 1) / √x
= (x - 1) / √x
= (3 + 2√2 - 1) / (√2 + 1)
= (2√2 + 2) / (√2 + 1)
= 2(√2 + 1) / (√2 + 1)
= 2
Another alternative :
(√x - 1/√x)²
= (√x)² - 2√x (1/√x) + (1/√x)²
= x - 2 + 1/x
= 3 + 2√2 - 2 + 1/(3 + 2√2)
= 2√2 + 1 + (3 - 2√2) / [(3 + 2√2)(3 - 2√2)]
= 2√2 + 1 + (3 - 2√2) / (3² - (2√2)²)
= 2√2 + 1 + (3 - 2√2) / (9 - 8)
= 2√2 + 1 + 3 - 2√2
= 4
So , √x - 1/√x = √4 = 2
(√x - 1/√x)²
= (√x)² - 2√x (1/√x) + (1/√x)²
= x - 2 + 1/x
= 3 + 2√2 - 2 + 1/(3 + 2√2)
= 2√2 + 1 + (3 - 2√2) / [(3 + 2√2)(3 - 2√2)]
= 2√2 + 1 + (3 - 2√2) / (3² - (2√2)²)
= 2√2 + 1 + (3 - 2√2) / (9 - 8)
= 2√2 + 1 + 3 - 2√2
= 4
So , √x - 1/√x = √4 = 2
thank u!
Welcome.
Something to note :
Since (√x - 1/√x)² = 4, you may be wondering if √x - 1/√x = ± √4 = ± 2
Why is it 2 and not - 2? Why is -2 rejected?
Because √x - 1/√x = (x - 1)/√x
= (3 + 2√2 - 1) / √(3 + 2√2)
= (2 + 2√2) / √(3 + 2√2)
Both the numerator and denominator are positive, so √x - 1/√x is positive.
So when we square root the 4, we take the positive square root.
Something to note :
Since (√x - 1/√x)² = 4, you may be wondering if √x - 1/√x = ± √4 = ± 2
Why is it 2 and not - 2? Why is -2 rejected?
Because √x - 1/√x = (x - 1)/√x
= (3 + 2√2 - 1) / √(3 + 2√2)
= (2 + 2√2) / √(3 + 2√2)
Both the numerator and denominator are positive, so √x - 1/√x is positive.
So when we square root the 4, we take the positive square root.
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