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Let-two-roots-of-the-equation-x-2-2mx-m-2-2m-3-0-is-For-lt-2-Find-2-Using-m-3-If-2-Solve-for-m-If-you-don-t-solve-it-I-will-force-you-to-solve-

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Question Number 215168 by MathematicalUser2357 last updated on 30/Dec/24
Let two roots of the equation x^2 +2mx+m^2 −2m+3=0 is α, β(For α<β), (2) Find (α−β)^2 Using m. (3) If β−α=2, Solve for m. If you don′t solve it, I will force you to solve.
$$\mathrm{Let}\:\mathrm{two}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} +\mathrm{2}{mx}+{m}^{\mathrm{2}} −\mathrm{2}{m}+\mathrm{3}=\mathrm{0}\:\mathrm{is}\:\alpha,\:\beta\left(\mathrm{For}\:\alpha<\beta\right), \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Find}\:\left(\alpha−\beta\right)^{\mathrm{2}} \:\mathrm{Using}\:{m}. \\ $$$$\left(\mathrm{3}\right)\:\mathrm{If}\:\beta−\alpha=\mathrm{2},\:\mathrm{Solve}\:\mathrm{for}\:{m}. \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{don}'\mathrm{t}\:\mathrm{solve}\:\mathrm{it},\:\mathrm{I}\:\mathrm{will}\:\mathrm{force}\:\mathrm{you}\:\mathrm{to}\:\mathrm{solve}. \\ $$
Answered by A5T last updated on 30/Dec/24
α+β=−2m; αβ=m^2 −2m+3 (α−β)^2 =(α+β)^2 −4αβ=8m−12 β−α=2⇒(α−β)^2 =4=8m−12⇒m=2
$$\alpha+\beta=−\mathrm{2m};\:\alpha\beta=\mathrm{m}^{\mathrm{2}} −\mathrm{2m}+\mathrm{3} \\ $$$$\left(\alpha−\beta\right)^{\mathrm{2}} =\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{4}\alpha\beta=\mathrm{8m}−\mathrm{12} \\ $$$$ \\ $$$$\beta−\alpha=\mathrm{2}\Rightarrow\left(\alpha−\beta\right)^{\mathrm{2}} =\mathrm{4}=\mathrm{8m}−\mathrm{12}\Rightarrow\mathrm{m}=\mathrm{2} \\ $$

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