FIND UNKNOWNS IN QUADRATIC EQUATION WITH ALPHA BETA

Find Unknowns in Quadratic Equation with Alpha Beta :

Here we are going to see some example problems of finding unknowns in quadratic equation with alpha beta.

Formulas of Alpha Beta in Quadratic Equation

(α2 + β2)  =  (α + β)- 2αβ

(α3 - β3)  =  (α - β)+ 3αβ(α - β)

(α4 + β4)  =  (α2β2)- 2α2β2

α - β  =  √[(α + β)- 4αβ]

Find Unknowns in Quadratic Equation with Alpha Beta - Questions

Question 1 :

If α, β are the roots of 7x2+ax+2=0 and if β − α = −13/ Find the values of a.

Solution :

7x+ ax + 2=0

a = 7, b = a and c = 2

Sum of roots (α + β)  =  -b/a  =  -a/7

Product of roots (α β)  =  c/a  =  2/7

β − α = −13/7

-(α - β)  =  -13/7

(α - β)  =  -13/7

(α + β)2 - 4αβ  =  -13/7

(α + β)2 - 4αβ  =  (-13/7)2

(-a/7)2 - 4(2/7)  =  169/49

(a2/49) - (8/7)  =  169/49

a2 - 56  =  169

a2  =  225

a  =  ±15

Hence the values of a are -15 and 15.

Question 2 :

If one root of the equation 2y2 − ay + 64 = 0 is twice the other then find the values of a.

Solution :

Let α and β are two roots.

α  =  2β, β = β

Sum of roots (α + β)  =  -b/a  =  -(-a)/2  =  a/2

Product of roots (α β)  =  c/a  =  64/2  =  32

α + β  =  a/2

2β + β  =  a/2

3β  =  a/2

β  =  a/6 ---(1)

2β(β)  =  32

2β =  32

β2  =  16

β  =  ±4

When β = 4

4 = a/6

a  =  24

when β = -4

-4  =  a/6

a  =  -24

Question 3 :

If one root of the equation 3x2 + kx + 81 = 0 (having real roots) is the square of the other then find k.

Solution :

α = β2

Sum of roots (α + β)  =  -b/a  =  -k/3

Product of roots (α β)  =  c/a  =  81/3  =  27

α β  =  27

β2β  =  27

β3  =  33

β  =  3

α + β  =  -k/3  --(1)

β2+ β  =  -k/3

32+ 3  =  -k/3

12  =  -k/3

-k  =  36

k = -36

Hence the value of k is -36.

After having gone through the stuff given above, we hope that the students would have understood, "Find Unknowns in Quadratic Equation with Alpha Beta". 

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