Whakaoti i te x^2-10x-400=0

$${ x }^{ 2 } -10x-400=0$$

Whakaoti mō x

$$x=5\sqrt{17}+5\approx 25.615528128$$

$$x=5-5\sqrt{17}\approx -15.615528128$$

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.

$$x^{2}-10x-400=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me -400 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-400\right)}}{2}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-400\right)}}{2}$$

Whakareatia -4 ki te -400.

$$x=\frac{-\left(-10\right)±\sqrt{100+1600}}{2}$$

Tāpiri 100 ki te 1600.

$$x=\frac{-\left(-10\right)±\sqrt{1700}}{2}$$

Tuhia te pūtakerua o te 1700.

$$x=\frac{-\left(-10\right)±10\sqrt{17}}{2}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±10\sqrt{17}}{2}$$

Nā, me whakaoti te whārite x=\frac{10±10\sqrt{17}}{2} ina he tāpiri te ±. Tāpiri 10 ki te 10\sqrt{17}.

$$x=\frac{10\sqrt{17}+10}{2}$$

Whakawehe 10+10\sqrt{17} ki te 2.

$$x=5\sqrt{17}+5$$

Nā, me whakaoti te whārite x=\frac{10±10\sqrt{17}}{2} ina he tango te ±. Tango 10\sqrt{17} mai i 10.

$$x=\frac{10-10\sqrt{17}}{2}$$

Whakawehe 10-10\sqrt{17} ki te 2.

$$x=5-5\sqrt{17}$$

Kua oti te whārite te whakatau.

$$x=5\sqrt{17}+5 x=5-5\sqrt{17}$$

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.

$$x^{2}-10x-400=0$$

Me tāpiri 400 ki ngā taha e rua o te whārite.

$$x^{2}-10x-400-\left(-400\right)=-\left(-400\right)$$

Mā te tango i te -400 i a ia ake anō ka toe ko te 0.

$$x^{2}-10x=-\left(-400\right)$$

Tango -400 mai i 0.

$$x^{2}-10x=400$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-10x+\left(-5\right)^{2}=400+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=400+25$$

Tāpiri 400 ki te 25.

$$x^{2}-10x+25=425$$

Tauwehea te x^{2}-10x+25. 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}.

$$\left(x-5\right)^{2}=425$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{425}$$

Whakarūnātia.

$$x-5=5\sqrt{17} x-5=-5\sqrt{17}$$

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

$$x=5\sqrt{17}+5 x=5-5\sqrt{17}$$