Whakaoti i te x^2-4x-5=11

$${ x }^{ 2 } -4x-5 = 11$$

Whakaoti mō x

$$x=2\sqrt{5}+2\approx 6.472135955$$

$$x=2-2\sqrt{5}\approx -2.472135955$$

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.

$$x^{2}-4x-5=11$$

Me tango 11 mai i ngā taha e rua o te whārite.

$$x^{2}-4x-5-11=11-11$$

Mā te tango i te 11 i a ia ake anō ka toe ko te 0.

$$x^{2}-4x-5-11=0$$

Tango 11 mai i -5.

$$x^{2}-4x-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -4 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -4.

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

Whakareatia -4 ki te -16.

$$x=\frac{-\left(-4\right)±\sqrt{16+64}}{2}$$

Tāpiri 16 ki te 64.

$$x=\frac{-\left(-4\right)±\sqrt{80}}{2}$$

Tuhia te pūtakerua o te 80.

$$x=\frac{-\left(-4\right)±4\sqrt{5}}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±4\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri 4 ki te 4\sqrt{5}.

$$x=\frac{4\sqrt{5}+4}{2}$$

Whakawehe 4+4\sqrt{5} ki te 2.

$$x=2\sqrt{5}+2$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{5}}{2} ina he tango te ±. Tango 4\sqrt{5} mai i 4.

$$x=\frac{4-4\sqrt{5}}{2}$$

Whakawehe 4-4\sqrt{5} ki te 2.

$$x=2-2\sqrt{5}$$

Kua oti te whārite te whakatau.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$

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.

$$x^{2}-4x-5=11$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$x^{2}-4x-5-\left(-5\right)=11-\left(-5\right)$$

Mā te tango i te -5 i a ia ake anō ka toe ko te 0.

$$x^{2}-4x=11-\left(-5\right)$$

Tango -5 mai i 11.

$$x^{2}-4x=16$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=16+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=16+4$$

Tāpiri 16 ki te 4.

$$x^{2}-4x+4=20$$

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}.

$$\left(x-2\right)^{2}=20$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{20}$$

Whakarūnātia.

$$x-2=2\sqrt{5} x-2=-2\sqrt{5}$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$