Whakaoti i te (9-x)(x-4)=3.x

$$( 9 - x ) ( x - 4 ) = 3 . x$$

Whakaoti mō x

$$x = \frac{\sqrt{13} + 13}{2} \approx 8.302775638$$

$$x = \frac{13 - \sqrt{13}}{2} \approx 4.697224362$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 9-x ki te x-4 ka whakakotahi i ngā kupu rite.

$$13x-36-x^{2}=3$$

Tangohia te 3 mai i ngā taha e rua.

$$13x-36-x^{2}-3=0$$

Tangohia te 3 i te -36, ka -39.

$$13x-39-x^{2}=0$$

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.

$$-x^{2}+13x-39=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 13 mō b, me -39 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\left(-39\right)}}{2\left(-1\right)}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\left(-1\right)\left(-39\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-13±\sqrt{169+4\left(-39\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -39.

$$x=\frac{-13±\sqrt{169-156}}{2\left(-1\right)}$$

Tāpiri 169 ki te -156.

$$x=\frac{-13±\sqrt{13}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-13±\sqrt{13}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{13}}{-2} ina he tāpiri te ±. Tāpiri -13 ki te \sqrt{13}.

$$x=\frac{\sqrt{13}-13}{-2}$$

Whakawehe -13+\sqrt{13} ki te -2.

$$x=\frac{13-\sqrt{13}}{2}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{13}}{-2} ina he tango te ±. Tango \sqrt{13} mai i -13.

$$x=\frac{-\sqrt{13}-13}{-2}$$

Whakawehe -13-\sqrt{13} ki te -2.

$$x=\frac{\sqrt{13}+13}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{13-\sqrt{13}}{2} x=\frac{\sqrt{13}+13}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 9-x ki te x-4 ka whakakotahi i ngā kupu rite.

$$13x-36-x^{2}=3$$

Me tāpiri te 36 ki ngā taha e rua.

$$13x-x^{2}=3+36$$

Tāpirihia te 3 ki te 36, ka 39.

$$13x-x^{2}=39$$

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.

$$-x^{2}+13x=39$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+13x}{-1}=\frac{39}{-1}$$

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

$$x^{2}+\frac{13}{-1}x=\frac{39}{-1}$$

Whakawehe 13 ki te -1.

$$x^{2}-13x=\frac{39}{-1}$$

Whakawehe 39 ki te -1.

$$x^{2}-13x=-39$$

Whakawehea te -13, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{2}. Nā, tāpiria te pūrua o te -\frac{13}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-39+\left(-\frac{13}{2}\right)^{2}$$

Pūruatia -\frac{13}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-13x+\frac{169}{4}=-39+\frac{169}{4}$$

Tāpiri -39 ki te \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{13}{4}$$

Tauwehea x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{13}{4}$$

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

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{13}{4}}$$

Whakarūnātia.

$$x-\frac{13}{2}=\frac{\sqrt{13}}{2} x-\frac{13}{2}=-\frac{\sqrt{13}}{2}$$

Me tāpiri \frac{13}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{13}+13}{2} x=\frac{13-\sqrt{13}}{2}$$