Whakaoti i te 2k^2-4k-15=0

$$2 k ^ { 2 } - 4 k - 15 = 0$$

Whakaoti mō k

$$k=\frac{\sqrt{34}}{2}+1\approx 3.915475947$$

$$k=-\frac{\sqrt{34}}{2}+1\approx -1.915475947$$

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.

$$2k^{2}-4k-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -4 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-15\right)}}{2\times 2}$$

Pūrua -4.

$$k=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-15\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$k=\frac{-\left(-4\right)±\sqrt{16-8\left(-15\right)}}{2\times 2}$$

Whakareatia -8 ki te -15.

$$k=\frac{-\left(-4\right)±\sqrt{16+120}}{2\times 2}$$

Tāpiri 16 ki te 120.

$$k=\frac{-\left(-4\right)±\sqrt{136}}{2\times 2}$$

Tuhia te pūtakerua o te 136.

$$k=\frac{-\left(-4\right)±2\sqrt{34}}{2\times 2}$$

Ko te tauaro o -4 ko 4.

$$k=\frac{4±2\sqrt{34}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$k=\frac{4±2\sqrt{34}}{4}$$

Nā, me whakaoti te whārite k=\frac{4±2\sqrt{34}}{4} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{34}.

$$k=\frac{2\sqrt{34}+4}{4}$$

Whakawehe 4+2\sqrt{34} ki te 4.

$$k=\frac{\sqrt{34}}{2}+1$$

Nā, me whakaoti te whārite k=\frac{4±2\sqrt{34}}{4} ina he tango te ±. Tango 2\sqrt{34} mai i 4.

$$k=\frac{4-2\sqrt{34}}{4}$$

Whakawehe 4-2\sqrt{34} ki te 4.

$$k=-\frac{\sqrt{34}}{2}+1$$

Kua oti te whārite te whakatau.

$$k=\frac{\sqrt{34}}{2}+1 k=-\frac{\sqrt{34}}{2}+1$$

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.

$$2k^{2}-4k-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$2k^{2}-4k-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$2k^{2}-4k=-\left(-15\right)$$

Tango -15 mai i 0.

$$2k^{2}-4k=15$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2k^{2}-4k}{2}=\frac{15}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$k^{2}+\left(-\frac{4}{2}\right)k=\frac{15}{2}$$

Whakawehe -4 ki te 2.

$$k^{2}-2k=\frac{15}{2}$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}-2k+1=\frac{15}{2}+1$$

Tāpiri \frac{15}{2} ki te 1.

$$k^{2}-2k+1=\frac{17}{2}$$

Tauwehea k^{2}-2k+1. 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}.

$$\left(k-1\right)^{2}=\frac{17}{2}$$

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

$$\sqrt{\left(k-1\right)^{2}}=\sqrt{\frac{17}{2}}$$

Whakarūnātia.

$$k-1=\frac{\sqrt{34}}{2} k-1=-\frac{\sqrt{34}}{2}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$k=\frac{\sqrt{34}}{2}+1 k=-\frac{\sqrt{34}}{2}+1$$