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If alpha and beta are the zeros pf p(x)= 3x^2+5x-2 then form a quadratic polynomial whose zeros are 2alpha and 2beta

If alpha and beta are the zeros pf p(x)= 3x^2+5x-2 then form a quadratic polynomial whose zeros are 2alpha and 2beta

Grade:10

8 Answers

Shreya D Nanda
74 Points
6 years ago
Hi Riya,
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`.
 
hope it is clear,,
sachin
85 Points
5 years ago
 
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`.
sachin
85 Points
5 years ago
 
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`.
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`.
sachin
85 Points
5 years ago
 
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`…,.,.
sachin
85 Points
5 years ago
 
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`.//,/,/,/
sachin
85 Points
5 years ago
 
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`.././/././.....
soumya ranjan prusty
23 Points
5 years ago
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`.
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`.
soumya ranjan prusty
23 Points
5 years ago
If \alpha , and \beta are roots of a quadratic equation, say, f\left ( x \right )=ax^2+ bx + c, then, we have
\alpha +\beta = -b/a  and \alpha\beta = c/a
So according too your question,
p\left ( x \right ) =3x^2 +5x-2
\therefore \alpha +\beta =-5/3  and   \therefore \alpha\beta =-2/3
 
To form an another equation, where 2\alpha , 2\beta are the roots then,
2\alpha + 2\beta= 2\left ( \alpha +\beta \right ) =2*-5/3=-10/3,   say      -b`/a`=-10/3 \Rightarrow a`=3 , b`=10
[2\alpha][2\beta] =4\alpha \beta =4*-2/3= -8/3    , say    c``/a`=-8/3 \Rightarrow a`=3 , c`=-8
 
Therefore,  the new quadratic polynomial, with 2\alpha , 2\beta as roots, is
p`\left ( x \right ) =a`x^2+b`x +c`.
\Rightarrow p`\left ( x \right ) =3x^2+10x -8`. ok....................................

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