Question

if alpha and beta are the zeros of x square - 6 X + K what is the value of k if 3 alpha + 2 B tech 20.
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Answer 1

Answer:-

Let alpha = p and beta = q.

Given Polynomial:

x² - 6x + k

Let a = 1 ; b = - 6 ; c = k

We know that,

sum of the zeroes = - b/a

→ p + q = - (- 6)/1

→ p + q = 6 -- equation (1)

Product of the zeroes = c/a

→ pq = k -- equation (2)

And also given that,

3p + 2q = 20 -- equation (3)

Multiplying equation (1) by 2 and subtracting equation (1) from (3) we get,

→ 3p + 2q - 2(p + q) = 20 - 2(6)

→ 3p + 2q - 2p - 2q = 20 - 12

→ p = 8

Substitute "p = 8" in equation (1)

→ p + q = 6

→ 8 + q = 6

→ q = 6 - 8

→ q = - 2

Putting the values of p & q in equation (2) we get,

→ (8)( - 2) = k

→ k = - 16

Hence, the value of k is - 16.

Answer 2

$\huge{\bold{\star{\fcolorbox{black}{lightgreen}{ANSWER}}}}⋆$

• The value of k = - 16.

$\huge{\bold{\star{\fcolorbox{black}{lightgreen}{EXPLANATION}}}}⋆$

$\blue{\bold{\underline{\underline{Let}}}}$

• Alpha = p.
• Beta = q.

$\blue{\bold{\underline{\underline{Given \: Polynomial}}}} </p><p>$

• x² - 6x + k

$\blue{\bold{\underline{\underline{Let}}}}$

• a = 1.
• b = -6.
• c = k.

$\blue{\bold{\underline{\underline{Formula}}}}$

${\boxed{\rm{Sum \: of \: the \: zeroes = \frac{- b}{a}}}}$

$\blue{\bold{\underline{\underline{Put \: the \: values}}}}$

$p + q = \frac{ - ( - 6)}{1}$

$\implies \: p + q = 6 \: ----- (i)$

$\blue{\bold{\underline{\underline{Formula}}}}$

${\boxed{\rm{Product \: of \: the \: zeroes = \frac{c}{a} }}}</p><p></p><p>$

$\blue{\bold{\underline{\underline{So}}}}$

$\implies \: pq = k ---- (ii)$

$\blue{\bold{\underline{\underline{Also \: Given}}}}$

$\implies 3p + 2q = 20 ---- (iii)$

$\blue{\bold{\underline{\underline{Multiplying \: equation \: (i) \: by \: (ii) \: and \: subtracting \: equation \: (i) \: from \: (iii)}}}}$

$3p + 2q -2(p + q) = 20 - 2(6)$

$=> 3p + 2q - 2p - 2q = 20 + 12$

$= > p = 8$

$\blue{\bold{\underline{\underline{Substitute \: (p = 8) \: in \: equation \: (i)}}}}$

$p + q = 6$

$\implies \: 8 + q = 6$

$\implies \: q = 6 - 8$

$\implies \: q = - 2$

$\blue{\bold{\underline{\underline{Putting \: the \: values \: of \: p \: and \: q \: equation \: (ii)}}}}</p><p>$

$(8)( - 2) = k$

$\implies \: k = - 16$

$\blue{\bold{\underline{\underline{ANSWER}}}}$

• The value of k = -16