FIND UNKNOWNS IN QUADRATIC EQUATION WITH ALPHA BETA

Question 1 :

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

Solution :

7x² + 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

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

Hence the value of k is -36.

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