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The roots of the equation x2/3 + x1/3 – 2 = 0 are ________.
- a)1, –8
- b)1, –2
- c)2/3, 1/3
- d)–2, –8
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The roots of the equation x2/3 + x1/3 – 2 = 0 are ________.a)1, ...
Given equation is x2/3 + x1/3 – 2 = 0
Let y = x1/3 ⇒ y2 + y – 2 = 0
⇒ y2 + 2y – y – 2 = 0 ⇒ (y – 1)(y + 2) = 0
⇒ y = 1 or y = –2 ⇒ x1/3 = 1 or x1/3 = –2
⇒ x = 1 or x = – 8
Let y = x1/3 ⇒ y2 + y – 2 = 0
⇒ y2 + 2y – y – 2 = 0 ⇒ (y – 1)(y + 2) = 0
⇒ y = 1 or y = –2 ⇒ x1/3 = 1 or x1/3 = –2
⇒ x = 1 or x = – 8
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The roots of the equation x2/3 + x1/3 – 2 = 0 are ________.a)1, ...
Understanding the Equation
The given equation is:
\[ x^{2/3} + x^{1/3} - 2 = 0 \]
To simplify and solve this equation, we introduce a substitution. Let:
\[ y = x^{1/3} \]
Thus, the equation can be rewritten as:
\[ y^2 + y - 2 = 0 \]
Factoring the Quadratic
Next, we will factor the quadratic equation:
1. Look for two numbers that multiply to \(-2\) (the constant term) and add to \(1\) (the coefficient of \(y\)).
2. The numbers \(2\) and \(-1\) satisfy this condition.
This allows us to factor the equation as follows:
\[ (y - 1)(y + 2) = 0 \]
Finding the Roots
Setting each factor to zero gives:
1. \( y - 1 = 0 \) → \( y = 1 \)
2. \( y + 2 = 0 \) → \( y = -2 \)
Back-Substituting for \(x\)
Now, we revert to our original variable \(x\):
1. From \( y = 1 \):
- \( x^{1/3} = 1 \) → \( x = 1^3 = 1 \)
2. From \( y = -2 \):
- \( x^{1/3} = -2 \) → \( x = (-2)^3 = -8 \)
Final Roots
Thus, the roots of the original equation \( x^{2/3} + x^{1/3} - 2 = 0 \) are:
- 1
- -8
Hence, the correct answer is option A: 1, -8.
The given equation is:
\[ x^{2/3} + x^{1/3} - 2 = 0 \]
To simplify and solve this equation, we introduce a substitution. Let:
\[ y = x^{1/3} \]
Thus, the equation can be rewritten as:
\[ y^2 + y - 2 = 0 \]
Factoring the Quadratic
Next, we will factor the quadratic equation:
1. Look for two numbers that multiply to \(-2\) (the constant term) and add to \(1\) (the coefficient of \(y\)).
2. The numbers \(2\) and \(-1\) satisfy this condition.
This allows us to factor the equation as follows:
\[ (y - 1)(y + 2) = 0 \]
Finding the Roots
Setting each factor to zero gives:
1. \( y - 1 = 0 \) → \( y = 1 \)
2. \( y + 2 = 0 \) → \( y = -2 \)
Back-Substituting for \(x\)
Now, we revert to our original variable \(x\):
1. From \( y = 1 \):
- \( x^{1/3} = 1 \) → \( x = 1^3 = 1 \)
2. From \( y = -2 \):
- \( x^{1/3} = -2 \) → \( x = (-2)^3 = -8 \)
Final Roots
Thus, the roots of the original equation \( x^{2/3} + x^{1/3} - 2 = 0 \) are:
- 1
- -8
Hence, the correct answer is option A: 1, -8.
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