Whakaoti i te (1/2(2x+2)(x+4))-(1/2(x+1)(x))=60

$$( \frac{ 1 }{ 2 } (2x+2)(x+4))-( \frac{ 1 }{ 2 } (x+1)(x))=60$$

Whakaoti mō x

$$x=-16$$

$$x=7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2} ki te 2x+2.

$$\left(x+1\right)\left(x+4\right)-\frac{1}{2}\left(x+1\right)x=60$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te x+4 ka whakakotahi i ngā kupu rite.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x=60$$

Tangohia te 60 mai i ngā taha e rua.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x-60=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{2} ki te x+1.

$$x^{2}+5x+4+\left(-\frac{1}{2}x-\frac{1}{2}\right)x-60=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{2}x-\frac{1}{2} ki te x.

$$x^{2}+5x+4-\frac{1}{2}x^{2}-\frac{1}{2}x-60=0$$

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

$$\frac{1}{2}x^{2}+5x+4-\frac{1}{2}x-60=0$$

Pahekotia te 5x me -\frac{1}{2}x, ka \frac{9}{2}x.

$$\frac{1}{2}x^{2}+\frac{9}{2}x+4-60=0$$

Tangohia te 60 i te 4, ka -56.

$$\frac{1}{2}x^{2}+\frac{9}{2}x-56=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{2} mō a, \frac{9}{2} mō b, me -56 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{2}±\sqrt{\left(\frac{9}{2}\right)^{2}-4\times \frac{1}{2}\left(-56\right)}}{2\times \frac{1}{2}}$$

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-4\times \frac{1}{2}\left(-56\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-2\left(-56\right)}}{2\times \frac{1}{2}}$$

Whakareatia -2 ki te -56.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}+112}}{2\times \frac{1}{2}}$$

Tāpiri \frac{81}{4} ki te 112.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{529}{4}}}{2\times \frac{1}{2}}$$

Tuhia te pūtakerua o te \frac{529}{4}.

$$x=\frac{-\frac{9}{2}±\frac{23}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{9}{2}±\frac{23}{2}}{1}$$

Nā, me whakaoti te whārite x=\frac{-\frac{9}{2}±\frac{23}{2}}{1} ina he tāpiri te ±. Tāpiri -\frac{9}{2} ki te \frac{23}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Whakawehe 7 ki te 1.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{-\frac{9}{2}±\frac{23}{2}}{1} ina he tango te ±. Tango \frac{23}{2} mai i -\frac{9}{2} mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{16}{1}$$

Whakawehe -16 ki te 1.

$$x=-16$$

Kua oti te whārite te whakatau.

$$x=7 x=-16$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2} ki te 2x+2.

$$\left(x+1\right)\left(x+4\right)-\frac{1}{2}\left(x+1\right)x=60$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te x+4 ka whakakotahi i ngā kupu rite.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x=60$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{2} ki te x+1.

$$x^{2}+5x+4+\left(-\frac{1}{2}x-\frac{1}{2}\right)x=60$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{2}x-\frac{1}{2} ki te x.

$$x^{2}+5x+4-\frac{1}{2}x^{2}-\frac{1}{2}x=60$$

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

$$\frac{1}{2}x^{2}+5x+4-\frac{1}{2}x=60$$

Pahekotia te 5x me -\frac{1}{2}x, ka \frac{9}{2}x.

$$\frac{1}{2}x^{2}+\frac{9}{2}x+4=60$$

Tangohia te 4 mai i ngā taha e rua.

$$\frac{1}{2}x^{2}+\frac{9}{2}x=60-4$$

Tangohia te 4 i te 60, ka 56.

$$\frac{1}{2}x^{2}+\frac{9}{2}x=56$$

Me whakarea ngā taha e rua ki te 2.

$$\frac{\frac{1}{2}x^{2}+\frac{9}{2}x}{\frac{1}{2}}=\frac{56}{\frac{1}{2}}$$

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

$$x^{2}+\frac{\frac{9}{2}}{\frac{1}{2}}x=\frac{56}{\frac{1}{2}}$$

Whakawehe \frac{9}{2} ki te \frac{1}{2} mā te whakarea \frac{9}{2} ki te tau huripoki o \frac{1}{2}.

$$x^{2}+9x=\frac{56}{\frac{1}{2}}$$

Whakawehe 56 ki te \frac{1}{2} mā te whakarea 56 ki te tau huripoki o \frac{1}{2}.

$$x^{2}+9x=112$$

Whakawehea te 9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{2}. Nā, tāpiria te pūrua o te \frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=112+\left(\frac{9}{2}\right)^{2}$$

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+9x+\frac{81}{4}=112+\frac{81}{4}$$

Tāpiri 112 ki te \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{529}{4}$$

Tauwehea x^{2}+9x+\frac{81}{4}. 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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{529}{4}$$

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

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Whakarūnātia.

$$x+\frac{9}{2}=\frac{23}{2} x+\frac{9}{2}=-\frac{23}{2}$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$x=7 x=-16$$