Whakaoti i te (2x+5)^2=(x+2)^2

$$( 2 x + 5 ) ^ { 2 } = ( x + 2 ) ^ { 2 }$$

Whakaoti mō x

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$x=-3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$4x^{2}+20x+25=\left(x+2\right)^{2}$$

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

$$4x^{2}+20x+25=x^{2}+4x+4$$

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

$$4x^{2}+20x+25-x^{2}=4x+4$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}+20x+25=4x+4$$

Tangohia te 4x mai i ngā taha e rua.

$$3x^{2}+20x+25-4x=4$$

Pahekotia te 20x me -4x, ka 16x.

$$3x^{2}+16x+25=4$$

Tangohia te 4 mai i ngā taha e rua.

$$3x^{2}+16x+25-4=0$$

Tangohia te 4 i te 25, ka 21.

$$3x^{2}+16x+21=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=16 ab=3\times 21=63$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 63.

$$1,63 3,21 7,9$$

Tātaihia te tapeke mō ia takirua.

$$1+63=64 3+21=24 7+9=16$$

Ko te otinga te takirua ka hoatu i te tapeke 16.

$$a=7 b=9$$

Tuhia anō te 3x^{2}+16x+21 hei \left(3x^{2}+7x\right)+\left(9x+21\right).

$$\left(3x^{2}+7x\right)+\left(9x+21\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(3x+7\right)+3\left(3x+7\right)$$

Whakatauwehea atu te kīanga pātahi 3x+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+7\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 3x+7=0 me te x+3=0.

$$x=-\frac{7}{3} x=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$4x^{2}+20x+25=\left(x+2\right)^{2}$$

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

$$4x^{2}+20x+25=x^{2}+4x+4$$

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

$$4x^{2}+20x+25-x^{2}=4x+4$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}+20x+25=4x+4$$

Tangohia te 4x mai i ngā taha e rua.

$$3x^{2}+20x+25-4x=4$$

Pahekotia te 20x me -4x, ka 16x.

$$3x^{2}+16x+25=4$$

Tangohia te 4 mai i ngā taha e rua.

$$3x^{2}+16x+25-4=0$$

Tangohia te 4 i te 25, ka 21.

$$3x^{2}+16x+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 16 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 3\times 21}}{2\times 3}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\times 3\times 21}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-16±\sqrt{256-12\times 21}}{2\times 3}$$

Whakareatia -12 ki te 21.

$$x=\frac{-16±\sqrt{256-252}}{2\times 3}$$

Tāpiri 256 ki te -252.

$$x=\frac{-16±\sqrt{4}}{2\times 3}$$

Tuhia te pūtakerua o te 4.

$$x=\frac{-16±2}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-16±2}{6}$$

Nā, me whakaoti te whārite x=\frac{-16±2}{6} ina he tāpiri te ±. Tāpiri -16 ki te 2.

$$x=-\frac{14}{6}$$

Whakahekea te hautanga \frac{-14}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{7}{3}$$

Nā, me whakaoti te whārite x=\frac{-16±2}{6} ina he tango te ±. Tango 2 mai i -16.

$$x=-\frac{18}{6}$$

Whakawehe -18 ki te 6.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=-\frac{7}{3} x=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$4x^{2}+20x+25=\left(x+2\right)^{2}$$

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

$$4x^{2}+20x+25=x^{2}+4x+4$$

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

$$4x^{2}+20x+25-x^{2}=4x+4$$

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}+20x+25=4x+4$$

Tangohia te 4x mai i ngā taha e rua.

$$3x^{2}+20x+25-4x=4$$

Pahekotia te 20x me -4x, ka 16x.

$$3x^{2}+16x+25=4$$

Tangohia te 25 mai i ngā taha e rua.

$$3x^{2}+16x=4-25$$

Tangohia te 25 i te 4, ka -21.

$$3x^{2}+16x=-21$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+16x}{3}=-\frac{21}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{16}{3}x=-\frac{21}{3}$$

Whakawehe -21 ki te 3.

$$x^{2}+\frac{16}{3}x=-7$$

Whakawehea te \frac{16}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{8}{3}. Nā, tāpiria te pūrua o te \frac{8}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=-7+\left(\frac{8}{3}\right)^{2}$$

Pūruatia \frac{8}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=-7+\frac{64}{9}$$

Tāpiri -7 ki te \frac{64}{9}.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{1}{9}$$

Tauwehea x^{2}+\frac{16}{3}x+\frac{64}{9}. 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}.

$$\left(x+\frac{8}{3}\right)^{2}=\frac{1}{9}$$

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

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Whakarūnātia.

$$x+\frac{8}{3}=\frac{1}{3} x+\frac{8}{3}=-\frac{1}{3}$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$x=-\frac{7}{3} x=-3$$