Whakaoti i te 2x/x^2+1=7/25

$$\frac { 2 x } { x ^ { 2 } + 1 } = \frac { 7 } { 25 }$$

Whakaoti mō x

$$x=7$$

$$x=\frac{1}{7}\approx 0.142857143$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me whakarea ngā taha e rua o te whārite ki te 25\left(x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+1,25.

$$25\times 2x=7\left(x^{2}+1\right)$$

Whakareatia te 25 ki te 2, ka 50.

$$50x=7\left(x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x^{2}+1.

$$50x=7x^{2}+7$$

Tangohia te 7x^{2} mai i ngā taha e rua.

$$50x-7x^{2}=7$$

Tangohia te 7 mai i ngā taha e rua.

$$50x-7x^{2}-7=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-7x^{2}+50x-7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -7x^{2}+ax+bx-7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=50 ab=-7\left(-7\right)=49$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 49.

$$1,49 7,7$$

Tātaihia te tapeke mō ia takirua.

$$1+49=50 7+7=14$$

Ko te otinga te takirua ka hoatu i te tapeke 50.

$$a=49 b=1$$

Tuhia anō te -7x^{2}+50x-7 hei \left(-7x^{2}+49x\right)+\left(x-7\right).

$$\left(-7x^{2}+49x\right)+\left(x-7\right)$$

Tauwehea te 7x i te tuatahi me te -1 i te rōpū tuarua.

$$7x\left(-x+7\right)-\left(-x+7\right)$$

Whakatauwehea atu te kīanga pātahi -x+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+7\right)\left(7x-1\right)$$

Hei kimi otinga whārite, me whakaoti te -x+7=0 me te 7x-1=0.

$$x=7 x=\frac{1}{7}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 25\left(x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+1,25.

$$25\times 2x=7\left(x^{2}+1\right)$$

Whakareatia te 25 ki te 2, ka 50.

$$50x=7\left(x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x^{2}+1.

$$50x=7x^{2}+7$$

Tangohia te 7x^{2} mai i ngā taha e rua.

$$50x-7x^{2}=7$$

Tangohia te 7 mai i ngā taha e rua.

$$50x-7x^{2}-7=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-7x^{2}+50x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -7 mō a, 50 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-7\right)\left(-7\right)}}{2\left(-7\right)}$$

Pūrua 50.

$$x=\frac{-50±\sqrt{2500-4\left(-7\right)\left(-7\right)}}{2\left(-7\right)}$$

Whakareatia -4 ki te -7.

$$x=\frac{-50±\sqrt{2500+28\left(-7\right)}}{2\left(-7\right)}$$

Whakareatia 28 ki te -7.

$$x=\frac{-50±\sqrt{2500-196}}{2\left(-7\right)}$$

Tāpiri 2500 ki te -196.

$$x=\frac{-50±\sqrt{2304}}{2\left(-7\right)}$$

Tuhia te pūtakerua o te 2304.

$$x=\frac{-50±48}{2\left(-7\right)}$$

Whakareatia 2 ki te -7.

$$x=\frac{-50±48}{-14}$$

Nā, me whakaoti te whārite x=\frac{-50±48}{-14} ina he tāpiri te ±. Tāpiri -50 ki te 48.

$$x=-\frac{2}{-14}$$

Whakahekea te hautanga \frac{-2}{-14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{7}$$

Nā, me whakaoti te whārite x=\frac{-50±48}{-14} ina he tango te ±. Tango 48 mai i -50.

$$x=-\frac{98}{-14}$$

Whakawehe -98 ki te -14.

$$x=7$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{7} x=7$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 25\left(x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+1,25.

$$25\times 2x=7\left(x^{2}+1\right)$$

Whakareatia te 25 ki te 2, ka 50.

$$50x=7\left(x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x^{2}+1.

$$50x=7x^{2}+7$$

Tangohia te 7x^{2} mai i ngā taha e rua.

$$50x-7x^{2}=7$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-7x^{2}+50x=7$$

Whakawehea ngā taha e rua ki te -7.

$$\frac{-7x^{2}+50x}{-7}=\frac{7}{-7}$$

Mā te whakawehe ki te -7 ka wetekia te whakareanga ki te -7.

$$x^{2}+\frac{50}{-7}x=\frac{7}{-7}$$

Whakawehe 50 ki te -7.

$$x^{2}-\frac{50}{7}x=\frac{7}{-7}$$

Whakawehe 7 ki te -7.

$$x^{2}-\frac{50}{7}x=-1$$

Whakawehea te -\frac{50}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{25}{7}. Nā, tāpiria te pūrua o te -\frac{25}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{50}{7}x+\left(-\frac{25}{7}\right)^{2}=-1+\left(-\frac{25}{7}\right)^{2}$$

Pūruatia -\frac{25}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{50}{7}x+\frac{625}{49}=-1+\frac{625}{49}$$

Tāpiri -1 ki te \frac{625}{49}.

$$x^{2}-\frac{50}{7}x+\frac{625}{49}=\frac{576}{49}$$

Tauwehea x^{2}-\frac{50}{7}x+\frac{625}{49}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{7}\right)^{2}=\frac{576}{49}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{25}{7}\right)^{2}}=\sqrt{\frac{576}{49}}$$

Whakarūnātia.

$$x-\frac{25}{7}=\frac{24}{7} x-\frac{25}{7}=-\frac{24}{7}$$

Me tāpiri \frac{25}{7} ki ngā taha e rua o te whārite.

$$x=7 x=\frac{1}{7}$$