Question

if alpha and beta are the zeroes of f (x) = x2-3x+2 find alpha2+beta2

Answer 1

f(x) = x² - 3x + 2 = 0
x² - x - 2x + 2 = 0
x(x-1) - 2(x-1) = 0
(x-1) (x-2) = 0
x = 1 ; x = 2

Therefore, 1 and 2 are zeroes of the polynomial.

Let α = 1 and ß = 2

=> sum of zeroes = 1+2
α+ß = 3

=> product of zeroes = (1) (2)
αß = 2

α² + ß² = (α+ß)² - 2αß
= (3)² - 2(2)
= 9 - 4
= 5

Answer 2

Hello !
your question already answered. But I want to answer your question because it is my favourite question. This question is from quadratic equation chapter.

given us,

f( x ) = x² - 3x + 2

we need to find alpha² + bita²

first we need to factorise ,

f( x ) = x² - 3x + 2

= x² - ( 2 + 1 )x + 2

= x² - 2x - x + 2

= x ( x - 2 ) - 1 ( x - 2 )

= ( x - 2 ) ( x - 1 )

so,

either,

x - 2 = 0

=> x = 2

or

x - 1 = 0

=> x = 1

here

2 and 1 are the zeros of this polynomial.

so, Let that

alpha = 1 and bita = 2

now

alpha + bita = 1 + 2 = 3

and

alpha × bita = 1 × 2 = 2

now,

alpha² + bita²

= ( alpha + bita )² - 2.alpha.bita [because ( a + b )² - 2ab = a² + b² + 2ab - 2ab = a² + b²]

= 3² - 2.2

= 9 - 4

= 5

.°. alpha² + bita² = 5

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