Question

find the zeroes of a quadratic polynomial 4x2-4x-3 and verify the relation between the zeroes and their cofficients​

Answer 1

$\Large\underline\mathrm\red{Given}$

Find the zeroes of a quadratic polynomial 4x² - 4x - 3 and verify the relation between the zeroes and their ccofficients

$\Large\underline\mathrm\red{Solution}$

• 4x² - 4x - 3

★ Solve it by splitting middle term

➠ 4x² - 4x - 3

➠ 4x² - 6x + 2x - 3

➠ 2x(2x - 3) + 1(2x - 3)

➠ (2x + 1)(2x - 3)

Either

➠ 2x + 1 = 0

➠ 2x = - 1

➠ x = -1/2

Or

➠ 2x - 3 = 0

➠ 2x = 3

➠ x = 3/2

Hence, -1/2 and 3/2 are the zeros of the given polynomial

$\Large\underline\mathrm\red{Verification}$

$\Large{\boxed{\bf{\blue{Sum\:of\:zeros}}}}$

★Add both the zeros

$\sf \implies\dfrac{-1}{2}+\dfrac{3}{2}=1=\dfrac{-(-4)}{4}=\dfrac{-b}{a} \\ \\ \sf \dfrac{-(Coefficient\:of\:x)}{Coefficient\:of\:x^2} \\ \\ \sf \Large{\boxed{\bf{\blue{Product\:of\:zeros}}}}$

★Multiply both the zeros

$\sf \implies\dfrac{-1}{2}\times\dfrac{3}{2}=\dfrac{-3}{4}=\dfrac{c}{a} \\ \\ \sf \dfrac{Constant\:term}{Coefficient\:of\:x^2}$

Answer 2

✯✯ QUESTION ✯✯

Find the zeroes of a quadratic polynomial 4x²-4x-3 and verify the relation between the zeroes and their cofficients..

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✰✰ ANSWER ✰✰

$\implies\tt{{4x}^{2}-4x-3}$

By Splitting Middle Term : -

$\implies\tt{{4x}^{2}-(6x-2x)-3}$

$\implies\tt{{4x}^{2}-6x+2x-3}$

$\implies\tt{2x(2x-2)+1(2x-3)}$

$\implies\tt{(2x+1)(2x-3)}$

• x=-1/2
• x=-1/2x = 3/2

➮So , -1/2 and 3/2 are the zeroes of polynomial 4x²-4x-3...

➥HERE : -

• a = 4
• b = -4
• c = -3

★Sum of Zeroes : -

$\implies\tt{\alpha+\beta=\dfrac{-b}{a}}$

$\implies\tt{\dfrac{-1}{2}+\dfrac{3}{2}=\dfrac{-(-4)}{4}}$

$\implies\tt{\dfrac{-1+3}{2}=\dfrac{4}{4}}$

$\implies\tt{\cancel\dfrac{2}{2}=\cancel\dfrac{4}{4}}$

$\implies\tt{1=1}$

$\red\longmapsto\:\large\underline{\boxed{\bf\green{L.H.S}\orange{=}\purple{R.H.S}}}$

★Product of Zeroes : -

$\implies\tt{\alpha\beta=\dfrac{c}{a}}$

$\implies\tt{\dfrac{-1}{2}\times\dfrac{3}{2}=\dfrac{-3}{4}}$

$\implies\tt{\dfrac{-3}{4}=\dfrac{-3}{4}}$

$\blue\longmapsto\:\large\underline{\boxed{\bf\pink{L.H.S}\orange{=}\red{R.H.S}}}$

✺ HENCE VERIFIED ✺