Whakaoti i te (16-x)(9-x)=112

$$( 16 - x ) ( 9 - x ) = 112$$

Whakaoti mō x

$$x = \frac{25 - \sqrt{497}}{2} \approx 1.353251595$$

$$x = \frac{\sqrt{497} + 25}{2} \approx 23.646748405$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$144-25x+x^{2}=112$$

Tangohia te 112 mai i ngā taha e rua.

$$144-25x+x^{2}-112=0$$

Tangohia te 112 i te 144, ka 32.

$$32-25x+x^{2}=0$$

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.

$$x^{2}-25x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -25 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 32}}{2}$$

Pūrua -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 32}}{2}$$

Whakareatia -4 ki te 32.

$$x=\frac{-\left(-25\right)±\sqrt{625-128}}{2}$$

Tāpiri 625 ki te -128.

$$x=\frac{-\left(-25\right)±\sqrt{497}}{2}$$

Ko te tauaro o -25 ko 25.

$$x=\frac{25±\sqrt{497}}{2}$$

Nā, me whakaoti te whārite x=\frac{25±\sqrt{497}}{2} ina he tāpiri te ±. Tāpiri 25 ki te \sqrt{497}.

$$x=\frac{\sqrt{497}+25}{2}$$

Nā, me whakaoti te whārite x=\frac{25±\sqrt{497}}{2} ina he tango te ±. Tango \sqrt{497} mai i 25.

$$x=\frac{25-\sqrt{497}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{497}+25}{2} x=\frac{25-\sqrt{497}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$144-25x+x^{2}=112$$

Tangohia te 144 mai i ngā taha e rua.

$$-25x+x^{2}=112-144$$

Tangohia te 144 i te 112, ka -32.

$$-25x+x^{2}=-32$$

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.

$$x^{2}-25x=-32$$

Whakawehea te -25, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{25}{2}. Nā, tāpiria te pūrua o te -\frac{25}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-32+\left(-\frac{25}{2}\right)^{2}$$

Pūruatia -\frac{25}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-25x+\frac{625}{4}=-32+\frac{625}{4}$$

Tāpiri -32 ki te \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{497}{4}$$

Tauwehea x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{497}{4}$$

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

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{497}{4}}$$

Whakarūnātia.

$$x-\frac{25}{2}=\frac{\sqrt{497}}{2} x-\frac{25}{2}=-\frac{\sqrt{497}}{2}$$

Me tāpiri \frac{25}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{497}+25}{2} x=\frac{25-\sqrt{497}}{2}$$