Question

# The quadratic equation whose roots are three times the roots of the equation $$2{ x }^{ 2 }+3x+5=0$$, is

A
2x2+9x+45=0
B
2x2+9x45=0
C
5x2+9x+45=0
D
2x29x+45=0
E
2x2+9x+49=0

Solution

## The correct option is B $$5{ x }^{ 2 }+9x+45=0$$Let the roots of the given equation $$2{ x }^{ 2 }+3x+5=0$$ be $$\alpha$$ and $$\beta$$.Then, $$\alpha +\beta =-\dfrac { 3 }{ 2 }$$ and $$\alpha \beta =\dfrac { 5 }{ 2 }$$Let the roots of required equation be $${ \alpha }^{ ' }$$ and $${ \beta }^{ ' }$$.It is given that,$${ \alpha }^{ ' }=3\alpha$$ and $${ \beta }^{ ' }=3\beta$$Now, $${ \alpha }^{ ' }+{ \beta }^{ ' }=3\alpha +3\beta$$$$=3\left( \alpha +\beta \right)$$$$=3\left( -\dfrac { 3 }{ 2 } \right) =-\dfrac { 9 }{ 2 }$$Also, $${ \alpha }^{ ' }\cdot { \beta }^{ ' }=\left( 3\alpha \right) \left( 3\beta \right) =9\alpha \beta$$$$=9\left( \dfrac { 5 }{ 2 } \right) =\dfrac { 45 }{ 2 }$$Hence, required equation is$${ x }^{ 2 }-\left( -\dfrac { 9 }{ 2 } \right) x+\dfrac { 45 }{ 2 } =0$$$$\Rightarrow 2{ x }^{ 2 }+9x+45=0$$Mathematics

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