Whakaoti i te (2x+3)*(x+5)=41

$$(2x+3) \times (x+5)=41$$

Whakaoti mō x

$$x = \frac{\sqrt{377} - 13}{4} \approx 1.60412196$$

$$x=\frac{-\sqrt{377}-13}{4}\approx -8.10412196$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+3 ki te x+5 ka whakakotahi i ngā kupu rite.

$$2x^{2}+13x+15=41$$

Tangohia te 41 mai i ngā taha e rua.

$$2x^{2}+13x+15-41=0$$

Tangohia te 41 i te 15, ka -26.

$$2x^{2}+13x-26=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 13 mō b, me -26 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 2\left(-26\right)}}{2\times 2}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\times 2\left(-26\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-13±\sqrt{169-8\left(-26\right)}}{2\times 2}$$

Whakareatia -8 ki te -26.

$$x=\frac{-13±\sqrt{169+208}}{2\times 2}$$

Tāpiri 169 ki te 208.

$$x=\frac{-13±\sqrt{377}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-13±\sqrt{377}}{4}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{377}}{4} ina he tāpiri te ±. Tāpiri -13 ki te \sqrt{377}.

$$x=\frac{\sqrt{377}-13}{4}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{377}}{4} ina he tango te ±. Tango \sqrt{377} mai i -13.

$$x=\frac{-\sqrt{377}-13}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{377}-13}{4} x=\frac{-\sqrt{377}-13}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+3 ki te x+5 ka whakakotahi i ngā kupu rite.

$$2x^{2}+13x+15=41$$

Tangohia te 15 mai i ngā taha e rua.

$$2x^{2}+13x=41-15$$

Tangohia te 15 i te 41, ka 26.

$$2x^{2}+13x=26$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+13x}{2}=\frac{26}{2}$$

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

$$x^{2}+\frac{13}{2}x=\frac{26}{2}$$

Whakawehe 26 ki te 2.

$$x^{2}+\frac{13}{2}x=13$$

Whakawehea te \frac{13}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{4}. Nā, tāpiria te pūrua o te \frac{13}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia \frac{13}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{13}{2}x+\frac{169}{16}=13+\frac{169}{16}$$

Tāpiri 13 ki te \frac{169}{16}.

$$x^{2}+\frac{13}{2}x+\frac{169}{16}=\frac{377}{16}$$

Tauwehea x^{2}+\frac{13}{2}x+\frac{169}{16}. 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}.

$$\left(x+\frac{13}{4}\right)^{2}=\frac{377}{16}$$

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

$$\sqrt{\left(x+\frac{13}{4}\right)^{2}}=\sqrt{\frac{377}{16}}$$

Whakarūnātia.

$$x+\frac{13}{4}=\frac{\sqrt{377}}{4} x+\frac{13}{4}=-\frac{\sqrt{377}}{4}$$

Me tango \frac{13}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{377}-13}{4} x=\frac{-\sqrt{377}-13}{4}$$