Whakaoti i te x^2+13x+15=0

$$x ^ { 2 } + 13 x + 15 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{109}-13}{2}\approx -1.279846746$$

$$x=\frac{-\sqrt{109}-13}{2}\approx -11.720153254$$

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.

$$x^{2}+13x+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 13 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 15}}{2}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\times 15}}{2}$$

Whakareatia -4 ki te 15.

$$x=\frac{-13±\sqrt{169-60}}{2}$$

Tāpiri 169 ki te -60.

$$x=\frac{-13±\sqrt{109}}{2}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{109}}{2} ina he tāpiri te ±. Tāpiri -13 ki te \sqrt{109}.

$$x=\frac{\sqrt{109}-13}{2}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{109}}{2} ina he tango te ±. Tango \sqrt{109} mai i -13.

$$x=\frac{-\sqrt{109}-13}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{109}-13}{2} x=\frac{-\sqrt{109}-13}{2}$$

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.

$$x^{2}+13x+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$x^{2}+13x+15-15=-15$$

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

$$x^{2}+13x=-15$$

Whakawehea te 13, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{2}. Nā, tāpiria te pūrua o te \frac{13}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+13x+\left(\frac{13}{2}\right)^{2}=-15+\left(\frac{13}{2}\right)^{2}$$

Pūruatia \frac{13}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+13x+\frac{169}{4}=-15+\frac{169}{4}$$

Tāpiri -15 ki te \frac{169}{4}.

$$x^{2}+13x+\frac{169}{4}=\frac{109}{4}$$

Tauwehea x^{2}+13x+\frac{169}{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}.

$$\left(x+\frac{13}{2}\right)^{2}=\frac{109}{4}$$

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

$$\sqrt{\left(x+\frac{13}{2}\right)^{2}}=\sqrt{\frac{109}{4}}$$

Whakarūnātia.

$$x+\frac{13}{2}=\frac{\sqrt{109}}{2} x+\frac{13}{2}=-\frac{\sqrt{109}}{2}$$

Me tango \frac{13}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{109}-13}{2} x=\frac{-\sqrt{109}-13}{2}$$