Whakaoti i te (x-15)(58-2x)=90

$$(x-15)(58-2x)=90$$

Whakaoti mō x

$$x=20$$

$$x=24$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-15 ki te 58-2x ka whakakotahi i ngā kupu rite.

$$88x-2x^{2}-870=90$$

Tangohia te 90 mai i ngā taha e rua.

$$88x-2x^{2}-870-90=0$$

Tangohia te 90 i te -870, ka -960.

$$88x-2x^{2}-960=0$$

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.

$$-2x^{2}+88x-960=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 88 mō b, me -960 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-88±\sqrt{88^{2}-4\left(-2\right)\left(-960\right)}}{2\left(-2\right)}$$

Pūrua 88.

$$x=\frac{-88±\sqrt{7744-4\left(-2\right)\left(-960\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-88±\sqrt{7744+8\left(-960\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -960.

$$x=\frac{-88±\sqrt{7744-7680}}{2\left(-2\right)}$$

Tāpiri 7744 ki te -7680.

$$x=\frac{-88±\sqrt{64}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-88±8}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-88±8}{-4}$$

Nā, me whakaoti te whārite x=\frac{-88±8}{-4} ina he tāpiri te ±. Tāpiri -88 ki te 8.

$$x=-\frac{80}{-4}$$

Whakawehe -80 ki te -4.

$$x=20$$

Nā, me whakaoti te whārite x=\frac{-88±8}{-4} ina he tango te ±. Tango 8 mai i -88.

$$x=-\frac{96}{-4}$$

Whakawehe -96 ki te -4.

$$x=24$$

Kua oti te whārite te whakatau.

$$x=20 x=24$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-15 ki te 58-2x ka whakakotahi i ngā kupu rite.

$$88x-2x^{2}-870=90$$

Me tāpiri te 870 ki ngā taha e rua.

$$88x-2x^{2}=90+870$$

Tāpirihia te 90 ki te 870, ka 960.

$$88x-2x^{2}=960$$

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.

$$-2x^{2}+88x=960$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+88x}{-2}=\frac{960}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$x^{2}+\frac{88}{-2}x=\frac{960}{-2}$$

Whakawehe 88 ki te -2.

$$x^{2}-44x=\frac{960}{-2}$$

Whakawehe 960 ki te -2.

$$x^{2}-44x=-480$$

Whakawehea te -44, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -22. Nā, tāpiria te pūrua o te -22 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-44x+\left(-22\right)^{2}=-480+\left(-22\right)^{2}$$

Pūrua -22.

$$x^{2}-44x+484=-480+484$$

Tāpiri -480 ki te 484.

$$x^{2}-44x+484=4$$

Tauwehea x^{2}-44x+484. 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}.

$$\left(x-22\right)^{2}=4$$

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

$$\sqrt{\left(x-22\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-22=2 x-22=-2$$

Me tāpiri 22 ki ngā taha e rua o te whārite.

$$x=24 x=20$$