Question

If alpha+beta are the zeroes of 4x^2+4x+1,then form a polynomial whose zeroes are -alpha and -beta

Answer 1

Hi Mate !


Here's the Solution :-

Given equation :- 4x² + 4x + 1


let's factorise it by middle term splitting !!

4x² + 4x + 1 = 0

4x² + 2x + 2x + 1 = 0

2x ( 2x + 1 ) + 1 ( 2x + 1 ) = 0

( 2x + 1 ) ( 2x + 1 ) = 0


• ( 2x + 1 ) = 0

x = ( - 1/2 )

• ( 2x + 1 ) = 0

x = ( - 1 /2 )


$let \: \: \alpha \: \: be \: \: \frac{ - 1}{2} \: \: and \: \: \beta \: \: \frac{ - 1}{2}$

• The new Equation have Zeros as :-

$- \alpha \: \: and \: \: - \beta$

$=)( - \alpha ) = - ( \frac{ - 1}{2} ) \\ = \frac{1}{2}$


$= > ( - \beta ) = - ( \frac{ - 1}{2} ) \\ = \frac{1}{2}$


• Sum of Zeros :-

$\frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1$


• Product of Zeros :-

$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$


♯ To form the quadratic equation we have formula as :-

x² - ( Sum of Zeros )x +( product of Zeros)

Putting value in the formula :-

${x}^{2} - x + \frac{1}{4} = 0$


$\frac{ 4{x}^{2} - 4x + 1}{4} = 0$


4x² - 4x + 1 = 0

is the required quadratic equation !!