Question

Find zeroes of quadratic pplynomial p(x) = 4√3x²+5x-2√3 and verify rrlationship between zeroes and its coefficients​ using alpha and beta

Answer 1

Answer:

√3/4 and -2/√3

Step-by-step explanation:

Refer attached image.

Answer 2

Given Equation

$\tt \to \: 4 \sqrt{3} {x}^{2} + 5x - 2 \sqrt{3} = 0$

To find

$\tt \to \: zeroes \: and \: verify \:$

Now Using Factorization method to find zeroes

$\tt \to \: 4 \sqrt{3} {x}^{2} + 5x - 2 \sqrt{3} = 0$

$\tt \to \: 4 \sqrt{3} {x}^{2} + 8x - 3x - 2 \sqrt{3} = 0$

$\tt \to \: 4x( \sqrt{3} x + 2) - \sqrt{3} ( \sqrt{3}x+ 2 ) = 0$

$\tt \to \: (4x - \sqrt{3} )( \sqrt{3} x + 2) = 0$

$\tt \to \: 4x - \sqrt{3} = 0 \: \: and \: \sqrt{3} x + 2 = 0$

$\tt \to 4x = \sqrt{3} \: \: and \: \sqrt{3}x = - 2$

$\tt \to \: x = \dfrac{ \sqrt{3} }{4} \: \: and \: x = \dfrac{ - 2}{ \sqrt{3} }$

we get

$\tt \to \alpha = \dfrac{ \sqrt{3} }{4} \: \: and \: \: \beta = \dfrac{ - 2}{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} } = \dfrac{ - 2 \sqrt{3} }{3}$

we have

$\tt \to \: a = 4 \sqrt{3} ,\: \: b = 5 \: and \: c = - 2 \sqrt{3}$

Sum of the zeroes are

$\tt \to \: ( \alpha + \beta ) = \dfrac{ - b}{a}$

$\tt \to \: \bigg( \dfrac{ \sqrt{3} }{4} - \dfrac{2 \sqrt{3} }{3} \bigg) = \dfrac{ - 5}{4 \sqrt{3} }$

$\tt \to \: \dfrac{3 \sqrt{3} - 8 \sqrt{3} }{12} = \dfrac{ - 5}{4 \sqrt{3} }$

$\tt \to \: \dfrac{ - 5 \sqrt{3} }{12} \times \dfrac{ \sqrt{3} }{ \sqrt{3} } = \dfrac{ - 5}{4 \sqrt{3} }$

$\tt \to \: \dfrac{ - 5 \times 3}{12 \sqrt{3} } = \dfrac{ - 5}{4 \sqrt{3} }$

$\tt \to \: \dfrac{ - 5}{4 \sqrt{3} } = \dfrac{ - 5}{4 \sqrt{3} }$

Product of zeroes

$\tt \to( \alpha \beta ) = \dfrac{ c}{a}$

$\tt \to \: \bigg( \dfrac{ \sqrt{3} }{4} \times \dfrac{ - 2}{ \sqrt{3} } \bigg) = \dfrac{ - 2 \sqrt{3} }{4 \sqrt{3} }$

$\tt \to \: \dfrac{ - 2 \sqrt{3} }{4 \sqrt{3} } = \dfrac{ - 1}{2}$

$\tt \to \dfrac{ - 1}{2} = \dfrac{ - 1}{2}$

Hence Proved