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Question Number 27258 by hasie09 last updated on 04/Jan/18

If 9^x −4×3^(x+2) +3^5 =0, then the solution set is

$$\mathrm{If}\:\:\mathrm{9}^{{x}} −\mathrm{4}×\mathrm{3}^{{x}+\mathrm{2}} +\mathrm{3}^{\mathrm{5}} =\mathrm{0},\:\mathrm{then}\:\mathrm{the}\:\mathrm{solution} \\ $$$$\mathrm{set}\:\mathrm{is} \\ $$

Commented by tawa tawa last updated on 04/Jan/18

(3^2 )^x − (4 × 3^x × 3^2 ) + 243 = 0 (3^x )^2 − (4 × 3^x × 9) + 243 = 0 Let 3^x = y ∴ y^2 −(4 × y × 9) + 243 = 0 ∴ y^2 −36y + 243 = 0 ∴ y^2 − 27y − 9y + 243 = 0 ∴ (y^2 − 27y) − (9y + 243) = 0 ∴ y(y − 27) − 9(y − 27) = 0 ∴ (y − 27)(y − 9) = 0 ∴ y − 27 = 0 or y − 9 = 0 ∴ y = 27 or y = 9 Remember: 3^x = y when y = 27 ∴ 3^x = 27 ∴ 3^x = 3^3 Since the base are equal ∴ x = 3 when y = 9 ∴ 3^x = 9 ∴ 3^x = 3^2 Since the base are equal ∴ x = 2 Therefore, x = 3 or x = 2

$$\left(\mathrm{3}^{\mathrm{2}} \right)^{\mathrm{x}} \:−\:\left(\mathrm{4}\:×\:\mathrm{3}^{\mathrm{x}} \:×\:\mathrm{3}^{\mathrm{2}} \right)\:+\:\mathrm{243}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{3}^{\mathrm{x}} \right)^{\mathrm{2}} \:−\:\left(\mathrm{4}\:×\:\mathrm{3}^{\mathrm{x}} \:×\:\mathrm{9}\right)\:+\:\mathrm{243}\:=\:\mathrm{0} \\ $$$$\mathrm{Let}\:\mathrm{3}^{\mathrm{x}} \:=\:\mathrm{y} \\ $$$$\therefore\:\mathrm{y}^{\mathrm{2}} \:−\left(\mathrm{4}\:×\:\mathrm{y}\:×\:\mathrm{9}\right)\:+\:\mathrm{243}\:=\:\mathrm{0} \\ $$$$\therefore\:\mathrm{y}^{\mathrm{2}} \:−\mathrm{36y}\:+\:\mathrm{243}\:=\:\mathrm{0} \\ $$$$\therefore\:\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{27y}\:−\:\mathrm{9y}\:+\:\mathrm{243}\:=\:\mathrm{0} \\ $$$$\therefore\:\left(\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{27y}\right)\:−\:\left(\mathrm{9y}\:+\:\mathrm{243}\right)\:=\:\mathrm{0} \\ $$$$\therefore\:\mathrm{y}\left(\mathrm{y}\:−\:\mathrm{27}\right)\:−\:\mathrm{9}\left(\mathrm{y}\:−\:\mathrm{27}\right)\:=\:\mathrm{0} \\ $$$$\therefore\:\left(\mathrm{y}\:−\:\mathrm{27}\right)\left(\mathrm{y}\:−\:\mathrm{9}\right)\:=\:\mathrm{0} \\ $$$$\therefore\:\mathrm{y}\:−\:\mathrm{27}\:=\:\mathrm{0}\:\mathrm{or}\:\mathrm{y}\:−\:\mathrm{9}\:=\:\mathrm{0} \\ $$$$\therefore\:\mathrm{y}\:=\:\mathrm{27}\:\mathrm{or}\:\mathrm{y}\:=\:\mathrm{9} \\ $$$$\mathrm{Remember}:\:\:\:\mathrm{3}^{\mathrm{x}} \:=\:\mathrm{y} \\ $$$$\mathrm{when}\:\:\:\mathrm{y}\:=\:\mathrm{27} \\ $$$$\therefore\:\:\:\mathrm{3}^{\mathrm{x}} \:=\:\mathrm{27} \\ $$$$\therefore\:\:\:\mathrm{3}^{\mathrm{x}} \:=\:\mathrm{3}^{\mathrm{3}} \\ $$$$\mathrm{Since}\:\mathrm{the}\:\mathrm{base}\:\mathrm{are}\:\mathrm{equal} \\ $$$$\therefore\:\:\:\:\mathrm{x}\:=\:\mathrm{3} \\ $$$$\mathrm{when}\:\mathrm{y}\:=\:\mathrm{9} \\ $$$$\therefore\:\:\:\mathrm{3}^{\mathrm{x}} \:=\:\mathrm{9} \\ $$$$\therefore\:\:\:\mathrm{3}^{\mathrm{x}} \:=\:\mathrm{3}^{\mathrm{2}} \\ $$$$\mathrm{Since}\:\mathrm{the}\:\mathrm{base}\:\mathrm{are}\:\mathrm{equal} \\ $$$$\therefore\:\:\:\:\mathrm{x}\:=\:\mathrm{2} \\ $$$$\mathrm{Therefore},\:\:\:\:\:\mathrm{x}\:=\:\mathrm{3}\:\:\:\mathrm{or}\:\:\:\:\mathrm{x}\:=\:\mathrm{2} \\ $$

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