The roots α and β of the quadratic equation x² - 5x + 3 (k-1) = 0 are such that α - β = 1.
Find the value of k.
Answers
Answer:
Q.E = x² - 5x + 3(k-1) = 0
Product of the root = αβ = -b/a
Sum of the roots = α + β = c/a
Where a = coefficient of x²
b= coefficient of x
c = constant
Note : When equated to zero
=> αβ = -b/a => -(-5)/1 => 5/1 = 5
=> α + β = c/a => 3(k-1)/1 => 3k - 3
Finding K:
Given :
α - β = 1
Solution:
- αβ = 5
- α + β = 3k -3
- α - β = 1
Adding equation 2 and 3:
=> α + β + α - β = 3k -3 +1
=> 2α = 3k -2
=> α = (3k -2) / 2
Subracting equation 2 and 3:
=> α + β - (α - β) = 3k -3 - 1
=> α + β - α + β = 3k -4
=>2β = 3k -4
=> β = (2k - 4)/2
=>β = k - 2
Substituting α and β in equation 1:
=> [(3k -2) / 2] (k-2) = 5
=> (3k -2)(k-2) / 2 = 5
=> (3k -2)(k-2) = 10
=> 3k² - 6k - 2k + 4 = 10
=> 3k²-8k +4 -10 = 0
=> 3k² - 8k -6 = 0
Solving quardratic equation using formula
=>k = (4 - √34)/3 and (4 + √34)/3
=> k ≈ - 0.61 and 3.27
→ Hey Mate,
→ Given Question:-
→ The roots α and β of the quadratic equation x² - 5x + 3 (k-1) = 0 are such that α - β = 1. Find the value of k.
→ To Find:-
→ Find the value of k?
→ Solution:-
→ x² -5x + 3( k - 1) = 0
→ α = 1
→ β = -5
→ c = 3 (k - 1)
→ α - β = 1
→ α - 1 = β
→ Sum of roots = α + β = -b / α
→ = α + α - 1 = - ( -5) / 1
→ = 2α - 1 = 5
→ = 2α = 6 (∵ 2 and 6 gets cancels)
→ so, = α = 3
→ β = α - 1
→ = 3 - 1
→ = 2
→ so , β = 2
→ Product of roots = αβ = c / a
→ = 3(2) = 3k - 3 / 1
→ = 6 = 3k - 3
→ = 6 + 3 = 3k
→ = 9 = 3k
→ = k = 3
→ ∴ The Value of k is 3.
→ More Information:-
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