Question

find the equation that formed by squaring each root of the equation x²+3x-2=0​

Answer 1

SOLUTION

TO DETERMINE

The quadratic equation that is formed by squaring each root of the equation

x² + 3x - 2 = 0

EVALUATION

Here the given Quadratic equation is

$\sf{ {x}^{2} + 3x - 2 = 0 \: \: \: \: ....(1) }$

Let $\sf{ \alpha \: \: and \: \: \beta }$

are the roots of the quadratic equation (1)

$\sf{ \therefore \: \: \alpha + \beta = - 3 \: \: \: and \: \: \alpha \beta = - 2}$

Now we have to find the quadratic equation whose roots are $\sf{ { \alpha }^{2} \: \: and \: \: { \beta }^{2} }$

Now

$\sf{ { \alpha }^{2} + { \beta }^{2} }$

$\sf{ = { (\alpha + \beta )}^{2} - 2 \alpha \beta }$

$\sf{ = { ( - 3 )}^{2} - 2 \times ( - 2)}$

$\sf{ = 9 + 4}$

$\sf{ = 13}$

Again

$\sf{ { \alpha }^{2} { \beta }^{2} }$

$= \sf{ { (\alpha \beta ) }^{2} }$

$= \sf{ { ( - 2 ) }^{2} }$

$= 4$

Hence the required Quadratic equation is

$\sf{ {x}^{2} - ( { \alpha }^{2} + { \beta }^{2} )x + { \alpha }^{2} { \beta }^{2} = 0 }$

$\implies \sf{ {x}^{2} - 13x + 4 = 0 }$

FINAL ANSWER

The required Quadratic equation is

$\sf{ {x}^{2} - 13x + 4 = 0 }$

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