. If α and β are zeroes of the quadratic polynomial x2 – 6x + a; find the value of ‘a’ if 3α + 2β = 20.
Answers
Answer: -16
Step-by-step explanation:
Given,
p(x): x² - 6x + a
Comparing this polynomial with standard form of quadratic polynomial
Ax² + Bx + C ,
A = 1
B = -6
C = a
Given that,
3α + 2β = 20
3α = 20 - 2β
3α = 2(10 - β)
α = 2/3(10 - β) ..........i)
We know that,
Sum of zeroes = -B/A
α + β = -(-6/1)
2/3(10 - β) + β = 6
20/3 - 2β/3 + β = 6
20/3 + β(1 - 2/3) = 6
20/3 + β(3 -2)/3 = 6
20/3 + β/3 = 6
20 + β = 6 × 3
20 + β = 18
β = 18 - 20
β = -2
Putting β = -2 in eq. i)
α = 2/3(10 + 2)
α = 2/3(12)
α = 2 × 4
α = 8
Now,
Product of zeroes = C/A
αβ = a/1
8 × (-2) = a
a = -16
Answer: -16
Step-by-step explanation:
Given,
p(x): x² - 6x + a
Comparing this polynomial with standard form of quadratic polynomial
Ax² + Bx + C ,
A = 1
B = -6
C = a
Given that,
3α + 2β = 20
3α = 20 - 2β
3α = 2(10 - β)
α = 2/3(10 - β) ..........i)
We know that,
Sum of zeroes = -B/A
α + β = -(-6/1)
2/3(10 - β) + β = 6
20/3 - 2β/3 + β = 6
20/3 + β(1 - 2/3) = 6
20/3 + β(3 -2)/3 = 6
20/3 + β/3 = 6
20 + β = 6 × 3
20 + β = 18
β = 18 - 20
β = -2
Putting β = -2 in eq. i)
α = 2/3(10 + 2)
α = 2/3(12)
α = 2 × 4
α = 8
Now,
Product of zeroes = C/A
αβ = a/1
8 × (-2) = a
a = -16
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