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Question Number 111724 by Aina Samuel Temidayo last updated on 04/Sep/20

How many real numbers x satisfy the equation 3^(2x+2) −3^(x+3) −3^x +3=0 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{x}\:\mathrm{satisfy}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{3}^{\mathrm{2x}+\mathrm{2}} −\mathrm{3}^{\mathrm{x}+\mathrm{3}} −\mathrm{3}^{\mathrm{x}} +\mathrm{3}=\mathrm{0}\:? \\ $$

Commented by mohammad17 last updated on 05/Sep/20

3^(2x) .3^2 −3^x .3^3 −3^x +3=0 set: k=3^x 9k^2 −28k+3=0 k=((−b∓(√(b^2 −4ac)))/(2a))⇒k=((28∓(√(784−108)))/(18)) k=((28∓(√(676)))/(18))⇒k=((28∓26)/(18)) either:k=(2/(18))⇒k=(1/9)⇒3^x =(1/9)⇒x=((ln((1/9)))/(ln(3)))⇒x=−2 or:k=((54)/(18))⇒k=3⇒3^x =3⇒x=1 ∴x=1 is true by:≪mohammad ≫

$$\mathrm{3}^{\mathrm{2}{x}} .\mathrm{3}^{\mathrm{2}} −\mathrm{3}^{{x}} .\mathrm{3}^{\mathrm{3}} −\mathrm{3}^{{x}} +\mathrm{3}=\mathrm{0} \\ $$$$ \\ $$$${set}:\:{k}=\mathrm{3}^{{x}} \\ $$$$ \\ $$$$\mathrm{9}{k}^{\mathrm{2}} −\mathrm{28}{k}+\mathrm{3}=\mathrm{0} \\ $$$$ \\ $$$${k}=\frac{−{b}\mp\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}}}{\mathrm{2}{a}}\Rightarrow{k}=\frac{\mathrm{28}\mp\sqrt{\mathrm{784}−\mathrm{108}}}{\mathrm{18}} \\ $$$$ \\ $$$${k}=\frac{\mathrm{28}\mp\sqrt{\mathrm{676}}}{\mathrm{18}}\Rightarrow{k}=\frac{\mathrm{28}\mp\mathrm{26}}{\mathrm{18}} \\ $$$$ \\ $$$${either}:{k}=\frac{\mathrm{2}}{\mathrm{18}}\Rightarrow{k}=\frac{\mathrm{1}}{\mathrm{9}}\Rightarrow\mathrm{3}^{{x}} =\frac{\mathrm{1}}{\mathrm{9}}\Rightarrow{x}=\frac{{ln}\left(\frac{\mathrm{1}}{\mathrm{9}}\right)}{{ln}\left(\mathrm{3}\right)}\Rightarrow{x}=−\mathrm{2}\: \\ $$$$ \\ $$$${or}:{k}=\frac{\mathrm{54}}{\mathrm{18}}\Rightarrow{k}=\mathrm{3}\Rightarrow\mathrm{3}^{{x}} =\mathrm{3}\Rightarrow{x}=\mathrm{1} \\ $$$$ \\ $$$$\therefore{x}=\mathrm{1}\:{is}\:{true} \\ $$$$ \\ $$$${by}:\ll{mohammad}\:\gg \\ $$

Commented by Aina Samuel Temidayo last updated on 05/Sep/20

Thanks.

$$\mathrm{Thanks}. \\ $$

Commented by Her_Majesty last updated on 05/Sep/20

why do you cancel x=−2? it′s a real number and it′s a solution.

$${why}\:{do}\:{you}\:{cancel}\:{x}=−\mathrm{2}?\:{it}'{s}\:{a}\:{real}\:{number} \\ $$$${and}\:{it}'{s}\:{a}\:{solution}. \\ $$

Commented by mohammad17 last updated on 05/Sep/20

im,sory sir am missed

$${im},{sory}\:{sir}\:{am}\:{missed} \\ $$

Commented by mohammad17 last updated on 05/Sep/20

you are welcome but sir x=−2 and x=1

$${you}\:{are}\:{welcome}\:{but}\:{sir}\:{x}=−\mathrm{2}\:{and}\:{x}=\mathrm{1} \\ $$

Commented by Her_Majesty last updated on 05/Sep/20

now it′s right!

$${now}\:{it}'{s}\:{right}! \\ $$

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