Question

if alpha and beta are the zeroes of the polynomial p(x) =x2-7x+12 then find the value of 1/alpha +1/beta​

Answer 1

Given :

If alpha and beta are the zeroes of the polynomial p(x) =x²-7x+12

To find :

• $\sf \dfrac{1}{\alpha}+\dfrac{1}{\beta}$

Solution :

$\\ \quad\bullet\:{\sf{\red{Given\: polynomial=x^2-7x+12}}}$

$\\ \qquad\quad\large\mathfrak{\underline{As\:we\:know\:that}}$

• Quadratic Polynomial : ax² + bx + c, where a ≠ 0

$\\\\{\large\purple\dagger}\:{\bf{\purple{Sum\:of\:roots=\dfrac{-(coefficient\:of\:x)}{(coefficient\:of\:x^2)}}}}$

$\implies\sf \alpha+\beta=\dfrac{(-b)}{a}$

$\implies\sf \alpha+\beta=\dfrac{-(-7)}{1}$

$\implies\sf \alpha +\beta=7$

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$\\{\large\purple\dagger}\:{\bf{\purple{Product\:of\:roots=\dfrac{(constant\:term)}{(coefficient\:of\:x^2)}}}}$

$\implies\sf \alpha\times \beta=\dfrac{c}{a}$

$\implies\sf \alpha\times \beta=\dfrac{12}{1}$

$\implies\sf \alpha \beta=12$

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• Value of 1/α + 1/β

$\\\implies\sf \dfrac{1}{\alpha}+\dfrac{1}{\beta}$

$\implies\sf \dfrac{\beta+\alpha}{\alpha \beta}$

$\implies\sf \dfrac{\alpha+\beta}{\alpha \beta}$

• Substitute the values of α+β & αβ

$\\\implies\sf \dfrac{7}{12}$

$\therefore{\underline{\boxed{\bf{Value\:of\: \dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{7}{12}}}}}$

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Answer 2

${\large{\bold{\rm{\underline{Question}}}}}$

★ If alpha and beta are the zeroes of the polynomial p(x) = x²-7x+12 then find the value of 1/alpha + 1/beta

${\large{\bold{\rm{\underline{Given \; that}}}}}$

${\sf{:\implies \alpha \: and \: \beta \: are \: zero \: of \: p(x) \: = \: x^{2} - 7x + 12}}$

${\large{\bold{\rm{\underline{To \; find}}}}}$

${\sf{:\implies The \: value \: of \: \dfrac{1}{\alpha} + \dfrac{1}{\beta}}}$

${\large{\bold{\rm{\underline{Solution}}}}}$

${\sf{:\implies The \: value \: of \: \dfrac{1}{\alpha} + \dfrac{1}{\beta} \: = 7/12}}$

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${\large{\bold{\rm{\underline{Using \; concept}}}}}$

${\sf{:\implies Sum \: of \: quadratic \: equation \: is \: given \: by \: what?}}$

${\sf{:\implies Product \: of \: quadratic \: equation \: is \: given \: by \: what?}}$

${\large{\bold{\rm{\underline{Using \; formula}}}}}$

${\sf{:\implies Sum \: of \: quadratic \: equation \: is \: given \: by \: \alpha + \beta \: = -b/a}}$

${\sf{:\implies Product \: of \: quadratic \: equation \: is \: given \: by \: \alpha \beta \: = c/a}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━

${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ As we know that sum of quadratic equation is given by α + β = -b/a. Henceforth,

${\sf{:\implies \alpha + \beta = -b/a}}$

${\sf{:\implies \alpha + \beta = -(-7)/1}}$

${\sf{:\implies \alpha + \beta = 7/1}}$

${\sf{:\implies \alpha + \beta = 7}}$ (1) Eq.

~ Now as e also know that product of quadratic equation is given by αβ = c/a. Henceforth,

${\sf{:\implies \alpha \beta = c/a}}$

${\sf{:\implies \alpha \beta = 12/1}}$

${\sf{:\implies \alpha \beta = 12}}$

~ Now let's find the value of 1/alpha + 1/beta.

${\sf{:\implies \beta + \alpha / \alpha \beta}}$

${\sf{:\implies 7/12}}$

${\small{\boxed{\boxed{\bf{7/12 \: is \: value \: of \: 1/alpha \: + \: 1/beta}}}}} \red \bigstar$

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${\large{\bold{\rm{\underline{Additional \; knowledge}}}}}$

Knowledge about Quadratic equations -

★ Sum of zeros of any quadratic equation is given by ➝ α+β = -b/a

★ Product of zeros of any quadratic equation is given by ➝ αβ = c/a

★ A quadratic equation have 2 roots

★ ax² + bx + c = 0 is the general form of quadratic equation