Question

If alpha and beta are roots of the quadratic equation y^2-2y-7=0 then find (1) alpha^2+beta^2 (2) alpha^3+beta^3

Answer 1

Answer:

Note;

Let a quadratic equation in variable y;

ay^2 + by + c = 0.

If alpha and beta are the roots of the quadratic equation,

Then,

(alpha + beta) = - b/a

and

alpha•beta = c/a.

Here,

The given quadratic equation is:

y^2 - 2y - 7 = 0

Clearly, we have

a =1

b = - 2

c = -7

Thus,

=> alpha + beta = - b/a

=> alpha + beta = - (-2/1) = 2 ------(1)

Also,

=> alpha•beta = c/a

=> alpha•beta = -7/1 = -7 ---------(2)

Now, we need to find;

1) alpha^2 + beta^2 = ?

2) alpha^3 + beta^3 = ?

Note :

i) x^2 + y^2 = (x+y)^2 - 2xy

ii) x^3 + y^3 = (x+y)(x^2 + y^2 - xy)

Now, using above identities,

We have;

1) alpha^2 + beta^2

= (alpha+beta)^2 - 2•alpha•beta

= (2)^2 - 2•(-7) {using eq-(1)&(2)}

= 4 + 14

= 18 ------(3)

Also,

2) alpha^3 + beta^3

= (alpha+beta)(alpha^2+beta^2

– alpha•beta)

= 2{18 - (-7)}

= 2{18 + 7}. {using eq-(1),(2)&(3)}

= 2•25

= 50.

Thus,

alpha^2 + beta^2 = 18

and

alpha^3 + beta^3 = 50.