Whakaoti i te 50left(1+xright)+left(1+xright)^2=450

$$50 \left( 1+x \right) + { \left(1+x \right) }^{ 2 } =450$$

Whakaoti mō x

$$x=5\sqrt{43}-26\approx 6.787192622$$

$$x=-5\sqrt{43}-26\approx -58.787192622$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 50 ki te 1+x.

$$50+50x+\left(1+x\right)^{2}=450$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$50+50x+1+2x+x^{2}=450$$

Tāpirihia te 50 ki te 1, ka 51.

$$51+50x+2x+x^{2}=450$$

Pahekotia te 50x me 2x, ka 52x.

$$51+52x+x^{2}=450$$

Tangohia te 450 mai i ngā taha e rua.

$$51+52x+x^{2}-450=0$$

Tangohia te 450 i te 51, ka -399.

$$-399+52x+x^{2}=0$$

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.

$$x^{2}+52x-399=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 52 mō b, me -399 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-52±\sqrt{52^{2}-4\left(-399\right)}}{2}$$

Pūrua 52.

$$x=\frac{-52±\sqrt{2704-4\left(-399\right)}}{2}$$

Whakareatia -4 ki te -399.

$$x=\frac{-52±\sqrt{2704+1596}}{2}$$

Tāpiri 2704 ki te 1596.

$$x=\frac{-52±\sqrt{4300}}{2}$$

Tuhia te pūtakerua o te 4300.

$$x=\frac{-52±10\sqrt{43}}{2}$$

Nā, me whakaoti te whārite x=\frac{-52±10\sqrt{43}}{2} ina he tāpiri te ±. Tāpiri -52 ki te 10\sqrt{43}.

$$x=\frac{10\sqrt{43}-52}{2}$$

Whakawehe -52+10\sqrt{43} ki te 2.

$$x=5\sqrt{43}-26$$

Nā, me whakaoti te whārite x=\frac{-52±10\sqrt{43}}{2} ina he tango te ±. Tango 10\sqrt{43} mai i -52.

$$x=\frac{-10\sqrt{43}-52}{2}$$

Whakawehe -52-10\sqrt{43} ki te 2.

$$x=-5\sqrt{43}-26$$

Kua oti te whārite te whakatau.

$$x=5\sqrt{43}-26 x=-5\sqrt{43}-26$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 50 ki te 1+x.

$$50+50x+\left(1+x\right)^{2}=450$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$50+50x+1+2x+x^{2}=450$$

Tāpirihia te 50 ki te 1, ka 51.

$$51+50x+2x+x^{2}=450$$

Pahekotia te 50x me 2x, ka 52x.

$$51+52x+x^{2}=450$$

Tangohia te 51 mai i ngā taha e rua.

$$52x+x^{2}=450-51$$

Tangohia te 51 i te 450, ka 399.

$$52x+x^{2}=399$$

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.

$$x^{2}+52x=399$$

Whakawehea te 52, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 26. Nā, tāpiria te pūrua o te 26 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+52x+26^{2}=399+26^{2}$$

Pūrua 26.

$$x^{2}+52x+676=399+676$$

Tāpiri 399 ki te 676.

$$x^{2}+52x+676=1075$$

Tauwehea x^{2}+52x+676. 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}.

$$\left(x+26\right)^{2}=1075$$

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

$$\sqrt{\left(x+26\right)^{2}}=\sqrt{1075}$$

Whakarūnātia.

$$x+26=5\sqrt{43} x+26=-5\sqrt{43}$$

Me tango 26 mai i ngā taha e rua o te whārite.

$$x=5\sqrt{43}-26 x=-5\sqrt{43}-26$$