Whakaoti i te 80*51=(80+x)(53-x)

$$80 \times 51 = ( 80 + x ) ( 53 - x )$$

Whakaoti mō x

$$x=-32$$

$$x=5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 80 ki te 51, ka 4080.

$$4080=\left(80+x\right)\left(53-x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 80+x ki te 53-x ka whakakotahi i ngā kupu rite.

$$4080=4240-27x-x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$4240-27x-x^{2}=4080$$

Tangohia te 4080 mai i ngā taha e rua.

$$4240-27x-x^{2}-4080=0$$

Tangohia te 4080 i te 4240, ka 160.

$$160-27x-x^{2}=0$$

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.

$$-x^{2}-27x+160=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -27 mō b, me 160 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\left(-1\right)\times 160}}{2\left(-1\right)}$$

Pūrua -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\left(-1\right)\times 160}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-27\right)±\sqrt{729+4\times 160}}{2\left(-1\right)}$$

Whakareatia 4 ki te 160.

$$x=\frac{-\left(-27\right)±\sqrt{729+640}}{2\left(-1\right)}$$

Tāpiri 729 ki te 640.

$$x=\frac{-\left(-27\right)±\sqrt{1369}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 1369.

$$x=\frac{-\left(-27\right)±37}{2\left(-1\right)}$$

Ko te tauaro o -27 ko 27.

$$x=\frac{27±37}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{27±37}{-2}$$

Nā, me whakaoti te whārite x=\frac{27±37}{-2} ina he tāpiri te ±. Tāpiri 27 ki te 37.

$$x=\frac{64}{-2}$$

Whakawehe 64 ki te -2.

$$x=-32$$

Nā, me whakaoti te whārite x=\frac{27±37}{-2} ina he tango te ±. Tango 37 mai i 27.

$$x=-\frac{10}{-2}$$

Whakawehe -10 ki te -2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=-32 x=5$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 80 ki te 51, ka 4080.

$$4080=\left(80+x\right)\left(53-x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 80+x ki te 53-x ka whakakotahi i ngā kupu rite.

$$4080=4240-27x-x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$4240-27x-x^{2}=4080$$

Tangohia te 4240 mai i ngā taha e rua.

$$-27x-x^{2}=4080-4240$$

Tangohia te 4240 i te 4080, ka -160.

$$-27x-x^{2}=-160$$

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.

$$-x^{2}-27x=-160$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-27x}{-1}=-\frac{160}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{27}{-1}\right)x=-\frac{160}{-1}$$

Whakawehe -27 ki te -1.

$$x^{2}+27x=-\frac{160}{-1}$$

Whakawehe -160 ki te -1.

$$x^{2}+27x=160$$

Whakawehea te 27, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{27}{2}. Nā, tāpiria te pūrua o te \frac{27}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+27x+\left(\frac{27}{2}\right)^{2}=160+\left(\frac{27}{2}\right)^{2}$$

Pūruatia \frac{27}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+27x+\frac{729}{4}=160+\frac{729}{4}$$

Tāpiri 160 ki te \frac{729}{4}.

$$x^{2}+27x+\frac{729}{4}=\frac{1369}{4}$$

Tauwehea x^{2}+27x+\frac{729}{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}.

$$\left(x+\frac{27}{2}\right)^{2}=\frac{1369}{4}$$

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

$$\sqrt{\left(x+\frac{27}{2}\right)^{2}}=\sqrt{\frac{1369}{4}}$$

Whakarūnātia.

$$x+\frac{27}{2}=\frac{37}{2} x+\frac{27}{2}=-\frac{37}{2}$$

Me tango \frac{27}{2} mai i ngā taha e rua o te whārite.

$$x=5 x=-32$$