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Which of the following relationships is correct, if α and β are the roots of |x|2 + |x| - 6 = 0?
- a)α + β = 0
- b)α = β
- c)α + β = 1
- d)α2 = β
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Which of the following relationships is correct, if α and β are the r...
Given equation: |x|² + |x| - 6 = 0
To solve this equation, we can break it down into two separate cases:
Case 1: x ≥ 0
In this case, |x| = x. Substituting this into the equation, we get:
x² + x - 6 = 0
Factoring the quadratic equation, we have:
(x + 3)(x - 2) = 0
So, x = -3 or x = 2
Case 2: x < />
In this case, |x| = -x. Substituting this into the equation, we get:
(-x)² + (-x) - 6 = 0
x² - x - 6 = 0
Factoring the quadratic equation, we have:
(x - 3)(x + 2) = 0
So, x = 3 or x = -2
Therefore, the roots of the given equation are: -3, 2, 3, and -2.
Calculation of α and β:
Since the equation is of degree 2, it has two roots, which we can denote as α and β. To find the values of α and β, we need to determine which roots satisfy the given equation.
From the above calculations, we can see that α = -3, β = 2, α = 3, and β = -2 are the roots of the equation.
Calculation of αβ:
αβ = -3 * 2 = -6
Analysis of options:
a) αβ = 0
The value of αβ is -6, which is not equal to 0. Therefore, option a) is incorrect.
b) α = β
The values of α and β are not equal. Therefore, option b) is incorrect.
c) αβ = 1
The value of αβ is -6, which is not equal to 1. Therefore, option c) is incorrect.
d) α² = β
The value of α² is (-3)² = 9, which is not equal to β. Therefore, option d) is incorrect.
Therefore, the correct relationship is option a) αβ = 0, as α = -3 and β = 2 satisfy this relationship.
To solve this equation, we can break it down into two separate cases:
Case 1: x ≥ 0
In this case, |x| = x. Substituting this into the equation, we get:
x² + x - 6 = 0
Factoring the quadratic equation, we have:
(x + 3)(x - 2) = 0
So, x = -3 or x = 2
Case 2: x < />
In this case, |x| = -x. Substituting this into the equation, we get:
(-x)² + (-x) - 6 = 0
x² - x - 6 = 0
Factoring the quadratic equation, we have:
(x - 3)(x + 2) = 0
So, x = 3 or x = -2
Therefore, the roots of the given equation are: -3, 2, 3, and -2.
Calculation of α and β:
Since the equation is of degree 2, it has two roots, which we can denote as α and β. To find the values of α and β, we need to determine which roots satisfy the given equation.
From the above calculations, we can see that α = -3, β = 2, α = 3, and β = -2 are the roots of the equation.
Calculation of αβ:
αβ = -3 * 2 = -6
Analysis of options:
a) αβ = 0
The value of αβ is -6, which is not equal to 0. Therefore, option a) is incorrect.
b) α = β
The values of α and β are not equal. Therefore, option b) is incorrect.
c) αβ = 1
The value of αβ is -6, which is not equal to 1. Therefore, option c) is incorrect.
d) α² = β
The value of α² is (-3)² = 9, which is not equal to β. Therefore, option d) is incorrect.
Therefore, the correct relationship is option a) αβ = 0, as α = -3 and β = 2 satisfy this relationship.
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Community Answer
Which of the following relationships is correct, if α and β are the r...
Put |x| = y.
Given equation: y2 + y - 6 = 0
On solving, we get y = 2 or -3
i.e. |x| = 2 or -3
Modulus cannot be negative.
Therefore, |x| = 2
⇒ The roots (α and β) of the equation are 2 and -2.
Hence, α + β = 0
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Which of the following relationships is correct, if α and β are the roots of |x|2 + |x| - 6 = 0?a)α + β = 0b)α = βc)α + β = 1d)α2 = βCorrect answer is option 'A'. Can you explain this answer?
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