Whakaoti i te 19-x^2-4x-1=0

$$19 - x ^ { 2 } - 4 x - 1 = 0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{22}-2\approx 2.69041576$$

$$x=-\left(\sqrt{22}+2\right)\approx -6.69041576$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$-x^{2}-4x+18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -4 mō b, me 18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 18}}{2\left(-1\right)}$$

Whakareatia 4 ki te 18.

$$x=\frac{-\left(-4\right)±\sqrt{16+72}}{2\left(-1\right)}$$

Tāpiri 16 ki te 72.

$$x=\frac{-\left(-4\right)±\sqrt{88}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 88.

$$x=\frac{-\left(-4\right)±2\sqrt{22}}{2\left(-1\right)}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{22}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{4±2\sqrt{22}}{-2}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{22}}{-2} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{22}.

$$x=\frac{2\sqrt{22}+4}{-2}$$

Whakawehe 4+2\sqrt{22} ki te -2.

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

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{22}}{-2} ina he tango te ±. Tango 2\sqrt{22} mai i 4.

$$x=\frac{4-2\sqrt{22}}{-2}$$

Whakawehe 4-2\sqrt{22} ki te -2.

$$x=\sqrt{22}-2$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$-x^{2}-4x+18=0$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-x^{2}-4x+18-18=-18$$

Mā te tango i te 18 i a ia ake anō ka toe ko te 0.

$$-x^{2}-4x=-18$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{18}{-1}$$

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

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{18}{-1}$$

Whakawehe -4 ki te -1.

$$x^{2}+4x=-\frac{18}{-1}$$

Whakawehe -18 ki te -1.

$$x^{2}+4x=18$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=18+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=18+4$$

Tāpiri 18 ki te 4.

$$x^{2}+4x+4=22$$

Tauwehea x^{2}+4x+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}.

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

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

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

Whakarūnātia.

$$x+2=\sqrt{22} x+2=-\sqrt{22}$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=\sqrt{22}-2 x=-\sqrt{22}-2$$

Whakaoti mō x

$$x=\sqrt{22}-2\approx 2.69041576$$

$$x=-\sqrt{22}-2\approx -6.69041576$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 1 i te 19, ka 18.

$$18-x^{2}-4x=0$$

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.

$$-x^{2}-4x+18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -4 mō b, me 18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 18}}{2\left(-1\right)}$$

Whakareatia 4 ki te 18.

$$x=\frac{-\left(-4\right)±\sqrt{16+72}}{2\left(-1\right)}$$

Tāpiri 16 ki te 72.

$$x=\frac{-\left(-4\right)±\sqrt{88}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 88.

$$x=\frac{-\left(-4\right)±2\sqrt{22}}{2\left(-1\right)}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{22}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{4±2\sqrt{22}}{-2}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{22}}{-2} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{22}.

$$x=\frac{2\sqrt{22}+4}{-2}$$

Whakawehe 4+2\sqrt{22} ki te -2.

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

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{22}}{-2} ina he tango te ±. Tango 2\sqrt{22} mai i 4.

$$x=\frac{4-2\sqrt{22}}{-2}$$

Whakawehe 4-2\sqrt{22} ki te -2.

$$x=\sqrt{22}-2$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 1 i te 19, ka 18.

$$18-x^{2}-4x=0$$

Tangohia te 18 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-x^{2}-4x=-18$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{18}{-1}$$

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

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{18}{-1}$$

Whakawehe -4 ki te -1.

$$x^{2}+4x=-\frac{18}{-1}$$

Whakawehe -18 ki te -1.

$$x^{2}+4x=18$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=18+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=18+4$$

Tāpiri 18 ki te 4.

$$x^{2}+4x+4=22$$

Tauwehea x^{2}+4x+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}.

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

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

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

Whakarūnātia.

$$x+2=\sqrt{22} x+2=-\sqrt{22}$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=\sqrt{22}-2 x=-\sqrt{22}-2$$