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Question Number 2191 by lakshaysethi039 last updated on 07/Nov/15

Find the number of quadratic equations having real roots and which do not change by squaring their roots.

$${Find}\:{the}\:{number}\:{of}\:{quadratic}\:{equations}\:{having}\:{real}\:{roots} \\ $$$${and}\:{which}\:{do}\:{not}\:{change}\:{by}\:{squaring}\:{their}\:{roots}. \\ $$$$ \\ $$

Answered by prakash jain last updated on 07/Nov/15

Eq: ax^2 +bx+c=0 Let roots be α,β Equation remain the same after roots are squared so same equation gives roots α^2 and β^2 . α=α^2 ,β=β^2 α=0,1 β=0,1 roots=(0,0)∨(1,0)∨(1,1) Three equations are x^2 =0 x^2 −x=0 x^2 −2x+1=0

$$\mathrm{Eq}:\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$$\mathrm{Let}\:\mathrm{roots}\:\mathrm{be}\:\alpha,\beta \\ $$$$\mathrm{Equation}\:\mathrm{remain}\:\mathrm{the}\:\mathrm{same}\:\mathrm{after}\:\mathrm{roots}\:\mathrm{are} \\ $$$$\mathrm{squared}\:\mathrm{so}\:\mathrm{same}\:\mathrm{equation}\:\mathrm{gives}\:\mathrm{roots}\:\alpha^{\mathrm{2}} \:\mathrm{and}\:\beta^{\mathrm{2}} . \\ $$$$\alpha=\alpha^{\mathrm{2}} ,\beta=\beta^{\mathrm{2}} \\ $$$$\alpha=\mathrm{0},\mathrm{1}\:\beta=\mathrm{0},\mathrm{1} \\ $$$$\mathrm{roots}=\left(\mathrm{0},\mathrm{0}\right)\vee\left(\mathrm{1},\mathrm{0}\right)\vee\left(\mathrm{1},\mathrm{1}\right) \\ $$$$\mathrm{Three}\:\mathrm{equations}\:\mathrm{are} \\ $$$${x}^{\mathrm{2}} =\mathrm{0} \\ $$$${x}^{\mathrm{2}} −{x}=\mathrm{0} \\ $$$${x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}=\mathrm{0} \\ $$

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