Check the solution to "If roots of the equation ax2+bx+c=0 are α,β,..."

Maeve Bowers

Answered question

2022-04-24

If roots of the equation

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

are

α, β

, find roots of equation

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

in terms of

α, β

smachttenbem

Beginner2022-04-25Added 18 answers

Let

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

and

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

we see that

g (x) = {(c + a)}^{2} \cdot \frac{1}{c} \cdot f (\frac{- c x}{c + a})

thus

g (x) = 0

implies

\frac{- c x}{c + a} = α, β

now use vieta and rearrange

Tyler Velasquez

Beginner2022-04-26Added 19 answers

Rewrite the equation

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

in the form

c {(1 + \frac{a}{c})}^{2} \frac{1}{x^{2}} - b (1 + \frac{a}{c}) \frac{1}{x} + a = 0

and compare with the given equation written in the form

\frac{c}{x^{2}} - \frac{b}{x} + a = 0

to establish the relationship between their roots

(1 + \frac{a}{c}) \frac{1}{x_{1}} = \frac{1}{α}, > > > > > (1 + \frac{a}{c}) \frac{1}{x_{2}} = \frac{1}{β}

which, with

\frac{c}{a} = α β

, leads to the roots

x_{1} = α + \frac{1}{β}, > > > > > x_{2} = β + \frac{1}{α}

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