Question

if alpha and beta are zeros of a quadratic polynomial 4X square + 4 x + 1 then form a quadratic polynomial whose zeros are 2 alpha and 2 Beta​

Answer 1

$\bold{\underline{\underline{\mathtt{\green{\bf{QUESTION:-}}}}}}$.

• if alpha and beta are zeros of a quadratic polynomial (4X² + 4 x + 1 )then form a quadratic polynomial whose zeros are 2 alpha and 2 Beta

_____________________

$\bold{\underline{\underline{\mathtt{\green{\bf{SOLUTION:-}}}}}}$.

$\bold{\underline{\underline{\:GIVEN\:HERE:-}}}$

• equation is 4x² + 4x +1 = 0

• alpha and beta are zeroes .

____________________

$\bold{\underline{\underline{\:FIND:HERE:-}}}$

• Find quadratic equation whose zeros are (2alpha ) and (2beta) .

______________________

$\bold{\underline{\underline{\:EXPLANATION:-}}}$

we know,

$\bold{\red{\boxed{\boxed{\:Sum\:of\:zeros\:=\frac{-(coefficient\:of\:x)}{(Coefficient\:of\:x^2)}}}}}$....(1)

$\implies\:(\alpha\:+\beta)\:=\cancel{\frac{-(4)}{4}}$

$\implies\:(\alpha\:+\beta)\:=\:-1$....(2)

____________________

Again,

$\bold{\red{\boxed{\boxed{\:Product\:of\:zeroes\:=\frac{(Constant\:part)}{(Coefficient\:of\:x^2)}}}}}$

$\implies\:(\alpha\beta)\:=\frac{1}{4}$...(3)

_____________________

Now, we using identity

$\bold{\red{\boxed{\boxed{\:(\alpha\:-\beta)\:=\sqrt{(\alpha\:+\beta)^2-4\alpha\beta}}}}}$

keep value by equation ,

$\implies\:(\alpha\:-\beta)\:=\sqrt{(-1)^2-\cancel{4}(\frac{1}{\cancel{4}})}$

$\implies\:(\alpha)\:-\beta)\:=\sqrt{(1-1)}$

$\implies\:(\alpha\:-\beta)\:=\:0$....(4)

_____________________

Addition of equation of (2) and (4)

we get,

$\implies\:(2\alpha)\:=\:-1$

$\bold{\boxed{\boxed{\alpha\:=\frac{(-1)}{2}}}}$....(5)

______________________

Now, keep value in (4) by (5).

$\implies\:(\frac{-1}{2}\:-\beta)\:=\:0$

$\bold{\boxed{\boxed{\:(\beta)\:=\frac{(-1)}{2}}}}$....(6)

______________________

Finally we found here ,

• $\alpha\:=\frac{-1}{2}$

• $\beta\:=\frac{-1}{2}$

_____________________

Now, we going to calculate quadratic equation , whose zeroes are $\:2\alpha\:and\:2\beta$.

Let,

• P and Q are zeroes of new quadratic equation

So,

First zeros will be ( P )

• $\:P\:=\:2\alpha$

• $\:P\:=\:\cancel{2}\times\frac{-1}{\cancel{2}}$

• $\:P\:=\:-1$....(7)

And,

Second zeroes will be (Q)

• $\:Q\:=\:2\beta$

• $\:Q\:=\:\cancel{2}\times\frac{-1}{\cancel{2}}$

• $\:Q\:=\:-1$. ...(8)

______________________

Now , we calculate

Sum Of P and Q :-

• (P+Q)=(-1)+(-1)= -2.

Product of P and Q :-

• (PQ)=(-1)(-1)=(1).

_____________________

$\bold{\underline{\underline{\mathtt{\green{\bf{\:Formula\:Of\:Quadratic\:Equation:-}}}}}}$.

$\bold{\red{\boxed{\boxed{\:X^2-X(P+Q)+PQ\:=\:0}}}}$

$\implies\:X^2-X(-2)+\:1\:=\:0$

So, Required. Quadratic equation

$\bold{\red{\boxed{\boxed{\:(X^2+2X+1)\:=\:0}}}}$

____________________

Answer 2

Answer:

please make my answer brainliest