Whakaoti i te (5-d)(5+10d)=(5+2d)^2

$$( 5 - d ) ( 5 + 10 d ) = ( 5 + 2 d ) ^ { 2 }$$

Whakaoti mō d

$$d = \frac{25}{14} = 1\frac{11}{14} \approx 1.785714286$$

$$d=0$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te āhuatanga tuaritanga hei whakarea te 5-d ki te 5+10d ka whakakotahi i ngā kupu rite.

$$25+45d-10d^{2}=\left(5+2d\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(5+2d\right)^{2}.

$$25+45d-10d^{2}=25+20d+4d^{2}$$

Tangohia te 25 mai i ngā taha e rua.

$$25+45d-10d^{2}-25=20d+4d^{2}$$

Tangohia te 25 i te 25, ka 0.

$$45d-10d^{2}=20d+4d^{2}$$

Tangohia te 20d mai i ngā taha e rua.

$$45d-10d^{2}-20d=4d^{2}$$

Pahekotia te 45d me -20d, ka 25d.

$$25d-10d^{2}=4d^{2}$$

Tangohia te 4d^{2} mai i ngā taha e rua.

$$25d-10d^{2}-4d^{2}=0$$

Pahekotia te -10d^{2} me -4d^{2}, ka -14d^{2}.

$$25d-14d^{2}=0$$

Tauwehea te d.

$$d\left(25-14d\right)=0$$

Hei kimi otinga whārite, me whakaoti te d=0 me te 25-14d=0.

$$d=0 d=\frac{25}{14}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 5-d ki te 5+10d ka whakakotahi i ngā kupu rite.

$$25+45d-10d^{2}=\left(5+2d\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(5+2d\right)^{2}.

$$25+45d-10d^{2}=25+20d+4d^{2}$$

Tangohia te 25 mai i ngā taha e rua.

$$25+45d-10d^{2}-25=20d+4d^{2}$$

Tangohia te 25 i te 25, ka 0.

$$45d-10d^{2}=20d+4d^{2}$$

Tangohia te 20d mai i ngā taha e rua.

$$45d-10d^{2}-20d=4d^{2}$$

Pahekotia te 45d me -20d, ka 25d.

$$25d-10d^{2}=4d^{2}$$

Tangohia te 4d^{2} mai i ngā taha e rua.

$$25d-10d^{2}-4d^{2}=0$$

Pahekotia te -10d^{2} me -4d^{2}, ka -14d^{2}.

$$25d-14d^{2}=0$$

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.

$$-14d^{2}+25d=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -14 mō a, 25 mō b, me 0 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-25±\sqrt{25^{2}}}{2\left(-14\right)}$$

Tuhia te pūtakerua o te 25^{2}.

$$d=\frac{-25±25}{2\left(-14\right)}$$

Whakareatia 2 ki te -14.

$$d=\frac{-25±25}{-28}$$

Nā, me whakaoti te whārite d=\frac{-25±25}{-28} ina he tāpiri te ±. Tāpiri -25 ki te 25.

$$d=\frac{0}{-28}$$

Whakawehe 0 ki te -28.

$$d=0$$

Nā, me whakaoti te whārite d=\frac{-25±25}{-28} ina he tango te ±. Tango 25 mai i -25.

$$d=-\frac{50}{-28}$$

Whakahekea te hautanga \frac{-50}{-28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$d=\frac{25}{14}$$

Kua oti te whārite te whakatau.

$$d=0 d=\frac{25}{14}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 5-d ki te 5+10d ka whakakotahi i ngā kupu rite.

$$25+45d-10d^{2}=\left(5+2d\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(5+2d\right)^{2}.

$$25+45d-10d^{2}=25+20d+4d^{2}$$

Tangohia te 20d mai i ngā taha e rua.

$$25+45d-10d^{2}-20d=25+4d^{2}$$

Pahekotia te 45d me -20d, ka 25d.

$$25+25d-10d^{2}=25+4d^{2}$$

Tangohia te 4d^{2} mai i ngā taha e rua.

$$25+25d-10d^{2}-4d^{2}=25$$

Pahekotia te -10d^{2} me -4d^{2}, ka -14d^{2}.

$$25+25d-14d^{2}=25$$

Tangohia te 25 mai i ngā taha e rua.

$$25d-14d^{2}=25-25$$

Tangohia te 25 i te 25, ka 0.

$$25d-14d^{2}=0$$

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.

$$-14d^{2}+25d=0$$

Whakawehea ngā taha e rua ki te -14.

$$\frac{-14d^{2}+25d}{-14}=\frac{0}{-14}$$

Mā te whakawehe ki te -14 ka wetekia te whakareanga ki te -14.

$$d^{2}+\frac{25}{-14}d=\frac{0}{-14}$$

Whakawehe 25 ki te -14.

$$d^{2}-\frac{25}{14}d=\frac{0}{-14}$$

Whakawehe 0 ki te -14.

$$d^{2}-\frac{25}{14}d=0$$

Whakawehea te -\frac{25}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{25}{28}. Nā, tāpiria te pūrua o te -\frac{25}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$d^{2}-\frac{25}{14}d+\left(-\frac{25}{28}\right)^{2}=\left(-\frac{25}{28}\right)^{2}$$

Pūruatia -\frac{25}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$d^{2}-\frac{25}{14}d+\frac{625}{784}=\frac{625}{784}$$

Tauwehea d^{2}-\frac{25}{14}d+\frac{625}{784}. 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}.

$$\left(d-\frac{25}{28}\right)^{2}=\frac{625}{784}$$

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

$$\sqrt{\left(d-\frac{25}{28}\right)^{2}}=\sqrt{\frac{625}{784}}$$

Whakarūnātia.

$$d-\frac{25}{28}=\frac{25}{28} d-\frac{25}{28}=-\frac{25}{28}$$

Me tāpiri \frac{25}{28} ki ngā taha e rua o te whārite.

$$d=\frac{25}{14} d=0$$