Whakaoti i te 4(2x-13)^2-9(2x-13)+2=0

$$4 ( 2 x - 13 ) ^ { 2 } - 9 ( 2 x - 13 ) + 2 = 0$$

Whakaoti mō x

$$x = \frac{53}{8} = 6\frac{5}{8} = 6.625$$

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

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-13\right)^{2}.

$$4\left(4x^{2}-52x+169\right)-9\left(2x-13\right)+2=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 4x^{2}-52x+169.

$$16x^{2}-208x+676-9\left(2x-13\right)+2=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -9 ki te 2x-13.

$$16x^{2}-208x+676-18x+117+2=0$$

Pahekotia te -208x me -18x, ka -226x.

$$16x^{2}-226x+676+117+2=0$$

Tāpirihia te 676 ki te 117, ka 793.

$$16x^{2}-226x+793+2=0$$

Tāpirihia te 793 ki te 2, ka 795.

$$16x^{2}-226x+795=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, -226 mō b, me 795 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-226\right)±\sqrt{\left(-226\right)^{2}-4\times 16\times 795}}{2\times 16}$$

Pūrua -226.

$$x=\frac{-\left(-226\right)±\sqrt{51076-4\times 16\times 795}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-\left(-226\right)±\sqrt{51076-64\times 795}}{2\times 16}$$

Whakareatia -64 ki te 795.

$$x=\frac{-\left(-226\right)±\sqrt{51076-50880}}{2\times 16}$$

Tāpiri 51076 ki te -50880.

$$x=\frac{-\left(-226\right)±\sqrt{196}}{2\times 16}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-\left(-226\right)±14}{2\times 16}$$

Ko te tauaro o -226 ko 226.

$$x=\frac{226±14}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{226±14}{32}$$

Nā, me whakaoti te whārite x=\frac{226±14}{32} ina he tāpiri te ±. Tāpiri 226 ki te 14.

$$x=\frac{240}{32}$$

Whakahekea te hautanga \frac{240}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$x=\frac{15}{2}$$

Nā, me whakaoti te whārite x=\frac{226±14}{32} ina he tango te ±. Tango 14 mai i 226.

$$x=\frac{212}{32}$$

Whakahekea te hautanga \frac{212}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{53}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{15}{2} x=\frac{53}{8}$$

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-13\right)^{2}.

$$4\left(4x^{2}-52x+169\right)-9\left(2x-13\right)+2=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 4x^{2}-52x+169.

$$16x^{2}-208x+676-9\left(2x-13\right)+2=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -9 ki te 2x-13.

$$16x^{2}-208x+676-18x+117+2=0$$

Pahekotia te -208x me -18x, ka -226x.

$$16x^{2}-226x+676+117+2=0$$

Tāpirihia te 676 ki te 117, ka 793.

$$16x^{2}-226x+793+2=0$$

Tāpirihia te 793 ki te 2, ka 795.

$$16x^{2}-226x+795=0$$

Tangohia te 795 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$16x^{2}-226x=-795$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}-226x}{16}=-\frac{795}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\left(-\frac{226}{16}\right)x=-\frac{795}{16}$$

Whakahekea te hautanga \frac{-226}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{113}{8}x=-\frac{795}{16}$$

Whakawehea te -\frac{113}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{113}{16}. Nā, tāpiria te pūrua o te -\frac{113}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{113}{8}x+\left(-\frac{113}{16}\right)^{2}=-\frac{795}{16}+\left(-\frac{113}{16}\right)^{2}$$

Pūruatia -\frac{113}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{113}{8}x+\frac{12769}{256}=-\frac{795}{16}+\frac{12769}{256}$$

Tāpiri -\frac{795}{16} ki te \frac{12769}{256} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{113}{8}x+\frac{12769}{256}=\frac{49}{256}$$

Tauwehea x^{2}-\frac{113}{8}x+\frac{12769}{256}. 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}.

$$\left(x-\frac{113}{16}\right)^{2}=\frac{49}{256}$$

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

$$\sqrt{\left(x-\frac{113}{16}\right)^{2}}=\sqrt{\frac{49}{256}}$$

Whakarūnātia.

$$x-\frac{113}{16}=\frac{7}{16} x-\frac{113}{16}=-\frac{7}{16}$$

Me tāpiri \frac{113}{16} ki ngā taha e rua o te whārite.

$$x=\frac{15}{2} x=\frac{53}{8}$$