Question

the sum and product of the zeroes of quadratic polynomial are -¹/2 and -3 respectively . what is quadratic equation​

Answer 1

Solution :-

Given ,

• sum of roots or zeroes = -½
• Product of roots or zeroes = -3

As we know that ,

• Quadratic polynomial :- x² - (α+β)x + αβ

Where ,

• α+β = sum of roots = - ½
• αβ = Product of roots = -3

Substituting the values we have ,

=> x² - (α+β)x + αβ = 0

=> x² - (-½)x + (-3) = 0

=> x² + x/2 - 3 = 0

=> (2x² + x)/2 - 3 = 0

=> (2x² + x - 6)/2 = 0

=> 2x² + x - 6 = 0

Hence , required quadratic equation is 2x² + x - 6 = 0

Answer 2

$\Large\bf\pink{Let,}$

• α & β are the zeroes of a quadratic polynomial.

$\Large{\underline{\bf{\color{cyan}GiVeN,}}}$

• The sum and product of the zeroes of a quadratic polynomial are -¹/₂ & -3.

$\longmapsto\:\:\bf\orange{\alpha\:+\:\beta\:=\:-\dfrac{1}{2}\:}$

$\longmapsto\:\:\bf\green{\alpha\:.\:\beta\:=\:-\:3\:}$

$\Large{\underline{\bf{\color{coral}To\:FiNd,}}}$

• The quadratic equation.

$\Large{\underline{\bf{\color{lime}CaLcUlAtIoN,}}}$

$\bf\blue{As\:we\:know \:that,}$

☆ The format of a quadratic equation is,

$\red\bigstar\:\:{\underline{\green{\boxed{\bf{\color{peru}x^2\:-\:(\alpha\:+\:\beta)\:x\:+\:\alpha\:\beta\:=\:0\:}}}}}$

$:\implies\:\:\bf{x^2\:-\:\Big(-\dfrac{1}{2}\Big)\:x\:+\:(-\:3)\:=\:0\:}$

$:\implies\:\:\bf{x^2\:+\:\dfrac{1}{2}\:x\:-\:3\:=\:0\:}$

✅ Multiple 2 in the above equation, we get

$:\implies\:\:\bf{2\times{x^2}\:+\:2\times{\dfrac{1}{2}\:x}\:-\:2\times{3}\:=\:0\:}$

$:\implies\:\:\bf\purple{2x^2\:+\:x\:-\:6\:=\:0\:}$

━─━─━─━─━─━─━─━─━─━─━─━─━─━─━

$\Large\bf\blue{Therefore,}$

The quadratic equation is 2x² + x - 6 = 0.