Question

the ratio of roots of the equation x^2+alphax+alpha+2=0 is 2.... find value of parameter alpha

Answer 1

Solution
________

Let the roots of equation be A B.

Then sum of roots = -(coefficient of x)/(coefficient of x²)

= A + B = -(alpha)/1 = A + B = -(alpha)........(1)

Product of roots = (constant term)/(coefficient of x²)

= AB = (alpha + 2)/1 = AB = alpha + 2 ........(2)

Also given in question A/B = 2

= A/B + B/A = 2 + 1/2 = (A² + B²)/AB = 5/2.......(3)

Now (A + B)² = A² + B² + 2AB

= (-alpha)² = A ² + B² + 2(alpha + 2) = (alpha)² = A² + B² + 2(alpha) + 4 (Putting values from equation (1) (2) )

= A² + B² = (alpha)² - 2(alpha) - 4

Putting this value in equation (3) we get

{(alpha)² - 2(alpha) - 4}/(alpha + 2) = 5/2

= 2{(alpha)² - 2(alpha) - 4)} = 5(alpha + 2) (Rearranging above equation)

= 2(alpha)² - 4(alpha) - 8 = 5(alpha) + 10

= 2(alpha)² - 4(alpha) - 5(alpha) - 8 - 10 = 0 (Rearranging above equation)

= 2(alpha)² - 9(alpha) - 18 = 0 = 2(alpha)² - 12(alpha) + 3(alpha) - 18 = 0

= 2 alpha (alpha - 6) + 3(alpha - 6) = 0

= {2(alpha) + 3} (alpha - 6) = 0

= either 2(alpha) + 3 = 0 OR alpha - 6 = 0

= either 2(alpha) = -3 OR alpha = 6

= either alpha = -3/2 OR alpha = 6

Answer 2

Answer:

Step-by-step explanation:

Let a and b be root of equation b and a =2

a=2b

sum of roots = −α

2b+b=−α

3b=−α

b= -3α

​

Product of roots = α+2

b×2b=α+2

2b^2=α+2

2b^2=−3b+2

2b^2+3b−2=0

2b^2 +4b−b−2=0

2b(b+2)−1(b+2)=0

b=−2 or b= 1/2

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If b=−2

α=−3×−2

α=6

If b=1/2

α=−3× 1/2

α=-3/2

Thank you