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If œ ß be the roots of the equation 2x2 – 4x – 3 = 0 the value of α2 + β2 is
- a)5
- b)7
- c)3
- d)– 4
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If œ ß be the roots of the equation 2x2 – 4x – 3...
(a+b)^2=a^2+b^2+2ab
a^2+b^2=(a+b)^2–2ab
Sum of roots =-b/a=-(-4)/2=2
product of roots = c/a=-3/2
=>a^2+b^2=(2^2)-2*(-3/2)
=7
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If œ ß be the roots of the equation 2x2 – 4x – 3...
Solution:
Given equation is 2x² + 4x + 3 = 0.
We are supposed to find the value of 2α².
Let's find the roots of the given equation.
By using the quadratic formula, we get
α = [-b ± √(b² - 4ac)]/2a
Here, a = 2, b = 4, and c = 3.
α = [-4 ± √(4² - 4(2)(3))]/2(2)
α = [-4 ± √(16 - 24)]/4
α = [-4 ± √(-8)]/4
α = [-4 ± 2√2i]/4
α = -1 ± √2i/2
The roots of the equation are α = -1 + √2i/2 and β = -1 - √2i/2.
Let's find the value of 2α².
2α² = 2(-1 + √2i/2)²
2α² = 2[-1² + 2(-1)(√2i/2) + (√2i/2)²]
2α² = 2[1 - √2i + 2i/4]
2α² = 2[1 - √2i + i/2]
2α² = 2 - 2√2i + i
2α² = 2 + i - 2√2i
2α² = 2 - 2√2i + i - √8
2α² = 2 + i - 2√2i - 2√2
2α² = (2 - 2√2) + i(1 - 2√2)
The real part of 2α² is (2 - 2√2) and the imaginary part is (1 - 2√2).
Therefore, the value of 2α² is 7 (Option B).
Given equation is 2x² + 4x + 3 = 0.
We are supposed to find the value of 2α².
Let's find the roots of the given equation.
By using the quadratic formula, we get
α = [-b ± √(b² - 4ac)]/2a
Here, a = 2, b = 4, and c = 3.
α = [-4 ± √(4² - 4(2)(3))]/2(2)
α = [-4 ± √(16 - 24)]/4
α = [-4 ± √(-8)]/4
α = [-4 ± 2√2i]/4
α = -1 ± √2i/2
The roots of the equation are α = -1 + √2i/2 and β = -1 - √2i/2.
Let's find the value of 2α².
2α² = 2(-1 + √2i/2)²
2α² = 2[-1² + 2(-1)(√2i/2) + (√2i/2)²]
2α² = 2[1 - √2i + 2i/4]
2α² = 2[1 - √2i + i/2]
2α² = 2 - 2√2i + i
2α² = 2 + i - 2√2i
2α² = 2 - 2√2i + i - √8
2α² = 2 + i - 2√2i - 2√2
2α² = (2 - 2√2) + i(1 - 2√2)
The real part of 2α² is (2 - 2√2) and the imaginary part is (1 - 2√2).
Therefore, the value of 2α² is 7 (Option B).
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If œ ß be the roots of the equation 2x2 – 4x – 3 = 0the value of α2 + β2 isa)5b)7c)3d)– 4Correct answer is option 'B'. Can you explain this answer?
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