Question

if alpha and beta are the roots of the equation (x^2+x-4=0) from the Quadratic Equation whose roots are alpha square and beta square (please reply fast please) ​

Answer 1

Given, α and β are roots of Quadratic equation x² +x -4 =0

Required to find : Quadratic equation whose roots are α²,β²

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We know that, In a Quadratic equation sum of roots and Product of roots is given by

$\: \: \: \boxed{ \mathfrak{ \alpha + \beta = \frac{ - b}{a} }}$

$\: \: \boxed{ \mathfrak{ \alpha \beta = \frac{ c}{a} }}$

x² + x -4 =0 Comparing with ax² + bx+ c = 0

a = 1 , b = 1 , c = -4

$\dashrightarrow \: \sf \: \alpha + \beta = \dfrac{ - 1}{1}$

$\dashrightarrow \: \sf \: \alpha \beta = \dfrac{ 4}{1}$

Now, We need the Quadratic equation whose roots are α²,β²

$\: \: \: \: \: \: \: \sf \underline{x {}^{2} - ( \alpha + \beta )x + \alpha \beta }$

α = α² , β= β²

$\: \: \: \:\: \sf \underline{ \pink{x {}^{2} - ( \alpha {}^{2} + \beta {}^{2} )x + \alpha {}^{2} \beta {}^{2} }}$

$\dashrightarrow \: \: \:\: \: \: \: \: \: \: \: \: \: \: \sf \: ( \alpha + \beta ) {}^{2} = (- 1) {}^{2}$

$\: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \dashrightarrow \: \sf \: \alpha {}^{2} + \beta {}^{2} + 2 \alpha \beta = 1$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \: \sf \: \alpha {}^{2} + \beta {}^{2} + 2(4) = 1$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \: \sf \: \alpha {}^{2} + \beta {}^{2} = 1 - 8$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \: \sf \: \boxed{ \pink{\alpha {}^{2} + \beta {}^{2} = - 7}}$

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

$\: \: \: \: \: \: \:\: \: \: \: \: \dashrightarrow \: \sf \: ( \alpha {}^{2} \beta {}^{2} ) = ( 4) {}^{2}$

$\: \: \: \: \: \: \dashrightarrow \: \sf \: \boxed{ \pink{( \alpha {}^{2} \beta {}^{2} ) = 16}}$

$\: \: \: \: \: \: \: \: \: \: \sf \underline{ \pink{x {}^{2} - ( \alpha {}^{2} + \beta {}^{2} )x + \alpha {}^{2} \beta {}^{2} }}$

$\: \: \: \: \: \: \: \: \sf \: x {}^{2} - ( - 7)x \: + 16$

$\: \: \: \: \: \: \: \: \: \: \sf \: x {}^{2} + 7x\: + 16$

$\: \: \: \sf \: \underline{The \: required \: quadratic \: eq {}^{n} \: whose \: roots \: are \: square \: of \: other \: is \: x {}^{2} + 7x + 16}$