Question

Find the zeros of the polynomial x² + 4x-5
and Verify the relation between the zeroes and the
co-efficient​

Answer 1

$\underline{\underline{\huge{\gray{\tt{\textbf Answer :-}}}}}$

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$\sf\large\bold{\orange{\underline{\blue{ Given :-}}}}$

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• A quadratic polynomial x² + 4x - 5

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$\sf\large\bold{\orange{\underline{\blue{ To \: Find :-}}}}$

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• Finding the zeroes of the given quadratic polynomial

• Verify the relation between the zeroes and the coefficient

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$\sf\large\bold{\orange{\underline{\blue{ Solution :-}}}}$

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Zeroes of quadratic polynomial

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➜ x² + 4x - 5

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By splitting the middle term method

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➜ x² + 4x - 5 = 0

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➜ x² + 5x - 1x - 5 = 0

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➜ x(x + 5)-1(x + 5) = 0

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➜ (x + 5)(x - 1) = 0

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• x = -5
• x = 1

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∴ The given quadratic polynomial has -5 & 1 as its zeroes

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Relationship between zeroes and coefficient

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$\underline{\bold{\texttt{For general quadratic polynomial :}}}$

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• For a quadratic polynomial ax² + bx + c

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$\underline{\bold{\texttt{Sum of zeroes :}}}$

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➠ $\bf \alpha + \beta = -\dfrac { b } { a }$ ⚊⚊⚊⚊ ⓵

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$\underline{\bold{\texttt{Product of zeroes :}}}$

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➠ $\bf \alpha \times \beta = \dfrac { c} { a }$ ⚊⚊⚊⚊ ⓶

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Where,

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• α = 1st zero

• β = 2nd zero

• a = Coefficient of x²

• b = Coefficient of x

• c = Constant term

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━━━━━━━━━━━━━━━━━━━━━━━━━

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$\underline{\bold{\texttt{For the given quadratic polynomial :}}}$

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• For the quadratic polynomial x² + 4x - 5

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$\underline{\bold{\texttt{Sum of zeroes :}}}$

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• α = -5

• β = 1

• a = 1

• b = 4

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⟮ Putting the above values in ⓵ ⟯

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➜ $\bf \alpha + \beta = -\dfrac { b } { a }$

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➜ $\sf -5 + 1 = -\dfrac { 4} { 1}$

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➜ -4 = -4

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Verified

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━━━━━━━━━━━━━━━━━━━━━━━━━

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$\underline{\bold{\texttt{Product of zeroes :}}}$

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• α = -5

• β = 1

• a = 1

• c = -5

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⟮ Putting the above values in ⓶ ⟯

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➜ $\bf \alpha \times \beta = \dfrac { c} { a }$

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➜ $\sf -5\times 1 = -\dfrac { -5} { 1}$

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➜ -5 = -5

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Verified

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