Whakaoti i te x^2-(7+x/2)((7+x/2)+x)=11

$${ x }^{ 2 } -( \frac{ 7+x }{ 2 } )(( \frac{ 7+x }{ 2 } )+x)=11$$

Whakaoti mō x

$$x=-3$$

$$x=31$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x^{2}-\left(7+x\right)\left(\frac{7+x}{2}+x\right)=22$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7+x ki te \frac{7+x}{2}+x.

$$2x^{2}-\left(7\times \frac{7+x}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Tuhia te 7\times \frac{7+x}{2} hei hautanga kotahi.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Tuhia te x\times \frac{7+x}{2} hei hautanga kotahi.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+\frac{x\left(7+x\right)}{2}+x^{2}\right)=22$$

Tā te mea he rite te tauraro o \frac{7\left(7+x\right)}{2} me \frac{x\left(7+x\right)}{2}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$2x^{2}-\left(\frac{7\left(7+x\right)+x\left(7+x\right)}{2}+7x+x^{2}\right)=22$$

Mahia ngā whakarea i roto o 7\left(7+x\right)+x\left(7+x\right).

$$2x^{2}-\left(\frac{49+7x+7x+x^{2}}{2}+7x+x^{2}\right)=22$$

Whakakotahitia ngā kupu rite i 49+7x+7x+x^{2}.

$$2x^{2}-\left(\frac{49+14x+x^{2}}{2}+7x+x^{2}\right)=22$$

Hei kimi i te tauaro o \frac{49+14x+x^{2}}{2}+7x+x^{2}, kimihia te tauaro o ia taurangi.

$$2x^{2}-\frac{49+14x+x^{2}}{2}-7x-x^{2}=22$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}-\frac{49+14x+x^{2}}{2}-7x=22$$

Whakawehea ia wā o 49+14x+x^{2} ki te 2, kia riro ko \frac{49}{2}+7x+\frac{1}{2}x^{2}.

$$x^{2}-\left(\frac{49}{2}+7x+\frac{1}{2}x^{2}\right)-7x=22$$

Hei kimi i te tauaro o \frac{49}{2}+7x+\frac{1}{2}x^{2}, kimihia te tauaro o ia taurangi.

$$x^{2}-\frac{49}{2}-7x-\frac{1}{2}x^{2}-7x=22$$

Pahekotia te x^{2} me -\frac{1}{2}x^{2}, ka \frac{1}{2}x^{2}.

$$\frac{1}{2}x^{2}-\frac{49}{2}-7x-7x=22$$

Pahekotia te -7x me -7x, ka -14x.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x=22$$

Tangohia te 22 mai i ngā taha e rua.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x-22=0$$

Tangohia te 22 i te -\frac{49}{2}, ka -\frac{93}{2}.

$$\frac{1}{2}x^{2}-\frac{93}{2}-14x=0$$

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.

$$\frac{1}{2}x^{2}-14x-\frac{93}{2}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{2} mō a, -14 mō b, me -\frac{93}{2} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times \frac{1}{2}\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times \frac{1}{2}\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

Whakareatia -4 ki te \frac{1}{2}.

$$x=\frac{-\left(-14\right)±\sqrt{196-2\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

Whakareatia -2 ki te -\frac{93}{2}.

$$x=\frac{-\left(-14\right)±\sqrt{196+93}}{2\times \frac{1}{2}}$$

Tāpiri 196 ki te 93.

$$x=\frac{-\left(-14\right)±\sqrt{289}}{2\times \frac{1}{2}}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-\left(-14\right)±17}{2\times \frac{1}{2}}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±17}{2\times \frac{1}{2}}$$

Whakareatia 2 ki te \frac{1}{2}.

$$x=\frac{14±17}{1}$$

Nā, me whakaoti te whārite x=\frac{14±17}{1} ina he tāpiri te ±. Tāpiri 14 ki te 17.

$$x=\frac{31}{1}$$

Whakawehe 31 ki te 1.

$$x=31$$

Nā, me whakaoti te whārite x=\frac{14±17}{1} ina he tango te ±. Tango 17 mai i 14.

$$x=-\frac{3}{1}$$

Whakawehe -3 ki te 1.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=31 x=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x^{2}-\left(7+x\right)\left(\frac{7+x}{2}+x\right)=22$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7+x ki te \frac{7+x}{2}+x.

$$2x^{2}-\left(7\times \frac{7+x}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Tuhia te 7\times \frac{7+x}{2} hei hautanga kotahi.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Tuhia te x\times \frac{7+x}{2} hei hautanga kotahi.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+\frac{x\left(7+x\right)}{2}+x^{2}\right)=22$$

Tā te mea he rite te tauraro o \frac{7\left(7+x\right)}{2} me \frac{x\left(7+x\right)}{2}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$2x^{2}-\left(\frac{7\left(7+x\right)+x\left(7+x\right)}{2}+7x+x^{2}\right)=22$$

Mahia ngā whakarea i roto o 7\left(7+x\right)+x\left(7+x\right).

$$2x^{2}-\left(\frac{49+7x+7x+x^{2}}{2}+7x+x^{2}\right)=22$$

Whakakotahitia ngā kupu rite i 49+7x+7x+x^{2}.

$$2x^{2}-\left(\frac{49+14x+x^{2}}{2}+7x+x^{2}\right)=22$$

Hei kimi i te tauaro o \frac{49+14x+x^{2}}{2}+7x+x^{2}, kimihia te tauaro o ia taurangi.

$$2x^{2}-\frac{49+14x+x^{2}}{2}-7x-x^{2}=22$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}-\frac{49+14x+x^{2}}{2}-7x=22$$

Whakawehea ia wā o 49+14x+x^{2} ki te 2, kia riro ko \frac{49}{2}+7x+\frac{1}{2}x^{2}.

$$x^{2}-\left(\frac{49}{2}+7x+\frac{1}{2}x^{2}\right)-7x=22$$

Hei kimi i te tauaro o \frac{49}{2}+7x+\frac{1}{2}x^{2}, kimihia te tauaro o ia taurangi.

$$x^{2}-\frac{49}{2}-7x-\frac{1}{2}x^{2}-7x=22$$

Pahekotia te x^{2} me -\frac{1}{2}x^{2}, ka \frac{1}{2}x^{2}.

$$\frac{1}{2}x^{2}-\frac{49}{2}-7x-7x=22$$

Pahekotia te -7x me -7x, ka -14x.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x=22$$

Me tāpiri te \frac{49}{2} ki ngā taha e rua.

$$\frac{1}{2}x^{2}-14x=22+\frac{49}{2}$$

Tāpirihia te 22 ki te \frac{49}{2}, ka \frac{93}{2}.

$$\frac{1}{2}x^{2}-14x=\frac{93}{2}$$

Me whakarea ngā taha e rua ki te 2.

$$\frac{\frac{1}{2}x^{2}-14x}{\frac{1}{2}}=\frac{\frac{93}{2}}{\frac{1}{2}}$$

Mā te whakawehe ki te \frac{1}{2} ka wetekia te whakareanga ki te \frac{1}{2}.

$$x^{2}+\left(-\frac{14}{\frac{1}{2}}\right)x=\frac{\frac{93}{2}}{\frac{1}{2}}$$

Whakawehe -14 ki te \frac{1}{2} mā te whakarea -14 ki te tau huripoki o \frac{1}{2}.

$$x^{2}-28x=\frac{\frac{93}{2}}{\frac{1}{2}}$$

Whakawehe \frac{93}{2} ki te \frac{1}{2} mā te whakarea \frac{93}{2} ki te tau huripoki o \frac{1}{2}.

$$x^{2}-28x=93$$

Whakawehea te -28, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -14. Nā, tāpiria te pūrua o te -14 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-28x+\left(-14\right)^{2}=93+\left(-14\right)^{2}$$

Pūrua -14.

$$x^{2}-28x+196=93+196$$

Tāpiri 93 ki te 196.

$$x^{2}-28x+196=289$$

Tauwehea x^{2}-28x+196. 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}.

$$\left(x-14\right)^{2}=289$$

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

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{289}$$

Whakarūnātia.

$$x-14=17 x-14=-17$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$x=31 x=-3$$