If roots of equation ax2+bx+c=0 is α,β find the roots of equation bcx2+a(bc+a2)x+ac2=0 in terms of α,β.

The given quadratic equation has the roots α and β. Then we have thatα+β=−ba and α⋅β=caThe new quadratic bc⋅x2+a(bc+a2)x+ac2=0Let the roots of the new quadratic be m,n. Then we have thatm+n=−a⋅bc+a2bc=−a(1+a2bc)=−a⋅{(1−1αβ(α+β))}and mn=ac2bc=ab⋅c=−a⋅(αβα+β)Hence the quadratic has becomex2+a⋅(1−1αβ(α+β))x−a⋅(αβα+β)=0I haven't found out any way to calculate and substitute a yet (it seems too that we can't determine the roots just by α and β).

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High School
Algebra
Algebra I
Quadratic function and equation

Kymani Shepherd

Answered question

2022-04-22

If roots of equation

a x^{2} + b x + c = 0

α, β

find the roots of equation

b c x^{2} + a (b c + a^{2}) x + a c^{2} = 0

in terms of

α, β

Answer & Explanation

Payton Cantrell

Beginner2022-04-23Added 15 answers

The given quadratic equation has the roots

α and β

. Then we have that

α + β = - \frac{b}{a} and α \cdot β = \frac{c}{a}

The new quadratic

b c \cdot x^{2} + a (b c + a^{2}) x + a c^{2} = 0

Let the roots of the new quadratic be m,n. Then we have that

m + n = - a \cdot \frac{b c + a^{2}}{b c} = - a (1 + \frac{a^{2}}{b c}) = - a \cdot {(1 - \frac{1}{α β (α + β)})}

and

m n = \frac{a c^{2}}{b c} = \frac{a}{b} \cdot c = - a \cdot (\frac{α β}{α + β})

Hence the quadratic has become

x^{2} + a \cdot (1 - \frac{1}{α β (α + β)}) x - a \cdot (\frac{α β}{α + β}) = 0

I haven't found out any way to calculate and substitute a yet (it seems too that we can't determine the roots just by

α and β

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