Whakaoti i te left(16-2xright)left(9-xright)=112

$$\left( 16-2x \right) \left( 9-x \right) = 112$$

Whakaoti mō x

$$x=1$$

$$x=16$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 16-2x ki te 9-x ka whakakotahi i ngā kupu rite.

$$144-34x+2x^{2}=112$$

Tangohia te 112 mai i ngā taha e rua.

$$144-34x+2x^{2}-112=0$$

Tangohia te 112 i te 144, ka 32.

$$32-34x+2x^{2}=0$$

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.

$$2x^{2}-34x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -34 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 2\times 32}}{2\times 2}$$

Pūrua -34.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 2\times 32}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-34\right)±\sqrt{1156-8\times 32}}{2\times 2}$$

Whakareatia -8 ki te 32.

$$x=\frac{-\left(-34\right)±\sqrt{1156-256}}{2\times 2}$$

Tāpiri 1156 ki te -256.

$$x=\frac{-\left(-34\right)±\sqrt{900}}{2\times 2}$$

Tuhia te pūtakerua o te 900.

$$x=\frac{-\left(-34\right)±30}{2\times 2}$$

Ko te tauaro o -34 ko 34.

$$x=\frac{34±30}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{34±30}{4}$$

Nā, me whakaoti te whārite x=\frac{34±30}{4} ina he tāpiri te ±. Tāpiri 34 ki te 30.

$$x=\frac{64}{4}$$

Whakawehe 64 ki te 4.

$$x=16$$

Nā, me whakaoti te whārite x=\frac{34±30}{4} ina he tango te ±. Tango 30 mai i 34.

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

Whakawehe 4 ki te 4.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=16 x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 16-2x ki te 9-x ka whakakotahi i ngā kupu rite.

$$144-34x+2x^{2}=112$$

Tangohia te 144 mai i ngā taha e rua.

$$-34x+2x^{2}=112-144$$

Tangohia te 144 i te 112, ka -32.

$$-34x+2x^{2}=-32$$

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.

$$2x^{2}-34x=-32$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-34x}{2}=-\frac{32}{2}$$

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

$$x^{2}+\left(-\frac{34}{2}\right)x=-\frac{32}{2}$$

Whakawehe -34 ki te 2.

$$x^{2}-17x=-\frac{32}{2}$$

Whakawehe -32 ki te 2.

$$x^{2}-17x=-16$$

Whakawehea te -17, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{17}{2}. Nā, tāpiria te pūrua o te -\frac{17}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-16+\left(-\frac{17}{2}\right)^{2}$$

Pūruatia -\frac{17}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-17x+\frac{289}{4}=-16+\frac{289}{4}$$

Tāpiri -16 ki te \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{225}{4}$$

Tauwehea te x^{2}-17x+\frac{289}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{225}{4}$$

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

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Whakarūnātia.

$$x-\frac{17}{2}=\frac{15}{2} x-\frac{17}{2}=-\frac{15}{2}$$

Me tāpiri \frac{17}{2} ki ngā taha e rua o te whārite.

$$x=16 x=1$$