Question

Q. Find the zeroes of the quadratic polynomial : x2 – 2x – 8 and verify the relationship between the zeroes and the coefficients.​

Answer 1

Answer:

$\large{\underbrace{\underline{\bf\bigstar\:Required\:Solution:-}}}$

» In this question, first we are supposed to find the zeroes of the quadratic equation given.

» To find the zeroes of the polynomial,we may use the quadratic formula or middle term splitting.

» After finding the zeroes, we will verify the relationship between the zeroes and the coefficients of the polynomial.

◆ Finding the zeroes of the polynomial :

» Here, we would us middle term splitting to find the zeroes.

$\colon \implies \: \bf {x}^{2} - 2x - 8 = 0$

$\colon \implies \bf \: {x}^{2} - 4x + 2x - 8 = 0$

$\colon \implies \bf \: x(x - 4) + 2(x - 4) = 0$

$\colon \implies \bf \: (x + 2)(x - 4) = 0$

So,

Either -

$\mapsto \sf \: x + 2 = 0 \\ \\ \dashrightarrow \boxed { \pink{ \bf{x = - 2}}}$

Or -

$\mapsto \sf \: x - 4 = 0 \\ \\ \dashrightarrow \boxed{ \bf { \pink{x = 4}}}$

$\therefore\;{\underline{\sf Zeroes ~of~the~Polynomial~are \bf{-2}~\sf and~\bf{4.}}}$

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◆ Verifying the Relationship between Zeroes and Coefficients :

» Standard form of a Quadratic Equation is -

$\odot \: \boxed{ \tt a {x}^{2} + bx + c}$

Compare it with given equation -

$\odot \: \boxed{ \tt {x}^{2} - 2x - 8 }$

So,

• a = 1
• b = -2
• c = -8

We know :

• Zeroes = -2 & 4

Now,

$\colon \displaystyle \leadsto \sf \: sum \: of \: zeroes = \bf\frac{ - b}{a}$

LHS :

$\colon \rarr \sf \: sum \: of \: zeroes = \bf - 2 + 4 \\ \\ \implies \boxed{ \bf \purple2}$

RHS :

$\displaystyle \colon \rarr \sf \: \frac{ - b}{a} = \bf \frac{ - ( - 2)}{1} \\ \\ \implies \boxed{ \bf \purple2}$

So,

LHS = RHS

Hence Verified.

And

$\displaystyle \colon \leadsto \sf \: product \: of \: zeroes = \bf \frac{c}{a}$

LHS :

$\colon \rarr \sf \: product \: of \: zeroes = \bf - 2 \times 4 \\ \\ \implies \boxed{ \bf \blue{ - 8}}$

RHS :

$\displaystyle \colon \rarr \sf \: \frac{c}{a} = \bf\frac{ - 8}{1} \\ \\ \implies \boxed{ \bf \blue{ - 8}}$

So,

LHS = RHS

Hence Verified.

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