Question

find zero following quadratic and verify the relationship between the zeros and Coefficient 4x2 +4√3x+3=0​

Answer 1

Question :

find zero following quadratic and verify the relationship between the zeros and Coefficient 4x2 +4√3x+3=0

To find :

Zeros of quadratic equations and verify its relationship between zeros and coefficient

Solution :

Applying quadratic formula to solve this question

a = 4 b = 4√3 c = 3

Substitute all the value of a, b and c

$\implies\sf \frac{-b±\sqrt{b^2-4ac}}{2a}$

=> -4√3 ± √(4√3)²- 4 × 4 × 3/2×4

=> -4√3 ± √48 - 48/8

=> -4√3 + √0/8, -4√3 - 0/8

=> -√3/2, -√3/2 are the zeros

Verification :

Sum of zeros

= -√3/2 + -√3/2 = -2√3/2 = √3

$\sf \large\frac{-coefficient\:of\:x}{coefficient\:of\:x^2}$

Product of zeros

= -√3/2×(-√3/2) = √9/4 = 3/4

$\sf \large\frac{constant\:term}{coefficient\:of\:x^2}$

Answer 2

$\huge{\underline{\underline{\frak{\red{Question}}}}}$

find the zeros of following quadratic equation and verify the relationship between the zeros and Coefficient 4x² +4√3x+3=0

$\huge{\underline{\underline{\frak{\red{Find\:out}}}}}$

Zeros and verify the relationship between zeros and coefficient

$\huge{\underline{\underline{\frak{\red{Solution}}}}}$

We can find zeros by two method

$\large{\boxed{\bf{Method:1}}}$

$\implies\sf 4x^2+4\sqrt{3}+3=0$

Splitting middle term

$\implies\sf 4x^2+2\sqrt{3}+2\sqrt{3}+3=0$

$\implies\sf 2x(2x+\sqrt{3})+\sqrt{3}(2x+\sqrt{3})=0$

$\implies\sf (2x+\sqrt{3})(2x+\sqrt{3})=0$

So,

$\sf 2x+\sqrt{3}=0$

$\sf 2x=-\sqrt{3}$

$\sf x=\frac{\sqrt{-3}}{2}$

$\large{\boxed{\bf{Method:2}}}$

Using quadratic formula to solve this question

$\implies\sf 4x^2+4\sqrt{3}+3=0$

a= 4 b=4√3 c= 3

Substitute all the value of a,b and c

$\implies\sf D=b^2+4ac$

$\implies\sf D=(4\sqrt{3})^2-4\times{4}\times{3}$

$\implies\sf D=48-48=0$

$\implies\sf x=\frac{-b\pm\sqrt{D}}{2a}$

$\implies\sf x=\frac{-4\sqrt{3}\pm\sqrt{D}}{2\times{4}}$

$\implies\sf x=\frac{-4\sqrt{3}+0}{8},\frac{-4\sqrt{3}-0}{8}$

$\implies\sf x=\frac{4\sqrt{3}}{4\times{2}},\frac{4\sqrt{3}}{4\times{2}}$

$\implies\sf x=\frac{\sqrt{-3}}{2},\frac{\sqrt{-3}}{2}$

$\huge{\underline{\underline{\frak{\red{Verification}}}}}$

$\large{\boxed{\bf{Sum\:of\:zeros}}}$

$\sf =\frac{\sqrt{-3}}{2}+\frac{\sqrt{-3}}{2}$

= √3+√3/2

= √3(1+1)/2 = 2√3/2 = √3

$\sf \large\frac{(-coefficient\:of\:x)}{(coefficient\:of\:x^2)}$

$\large{\boxed{\bf{Product\:of\:zeros}}}$

$\sf =\frac{\sqrt{-3}}{2}\times\frac{\sqrt{-3}}{2}$

$\sf =\large\frac{3}{4}$

$\sf \large\frac{(Constant\:term)}{(coefficient\:of\:x^2)}$