Whakaoti i te (3x-4)^2-(x+3)^2=0

$$( 3 x - 4 ) ^ { 2 } - ( x + 3 ) ^ { 2 } = 0$$

Whakaoti mō x

$$x=\frac{1}{4}=0.25$$

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3x-4\right)^{2}.

$$9x^{2}-24x+16-\left(x+3\right)^{2}=0$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$9x^{2}-24x+16-\left(x^{2}+6x+9\right)=0$$

Hei kimi i te tauaro o x^{2}+6x+9, kimihia te tauaro o ia taurangi.

$$9x^{2}-24x+16-x^{2}-6x-9=0$$

Pahekotia te 9x^{2} me -x^{2}, ka 8x^{2}.

$$8x^{2}-24x+16-6x-9=0$$

Pahekotia te -24x me -6x, ka -30x.

$$8x^{2}-30x+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$8x^{2}-30x+7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 8x^{2}+ax+bx+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-30 ab=8\times 7=56$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 56.

$$-1,-56 -2,-28 -4,-14 -7,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-56=-57 -2-28=-30 -4-14=-18 -7-8=-15$$

Ko te otinga te takirua ka hoatu i te tapeke -30.

$$a=-28 b=-2$$

Tuhia anō te 8x^{2}-30x+7 hei \left(8x^{2}-28x\right)+\left(-2x+7\right).

$$\left(8x^{2}-28x\right)+\left(-2x+7\right)$$

Tauwehea te 4x i te tuatahi me te -1 i te rōpū tuarua.

$$4x\left(2x-7\right)-\left(2x-7\right)$$

Whakatauwehea atu te kīanga pātahi 2x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-7\right)\left(4x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-7=0 me te 4x-1=0.

$$x=\frac{7}{2} x=\frac{1}{4}$$

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(3x-4\right)^{2}.

$$9x^{2}-24x+16-\left(x+3\right)^{2}=0$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$9x^{2}-24x+16-\left(x^{2}+6x+9\right)=0$$

Hei kimi i te tauaro o x^{2}+6x+9, kimihia te tauaro o ia taurangi.

$$9x^{2}-24x+16-x^{2}-6x-9=0$$

Pahekotia te 9x^{2} me -x^{2}, ka 8x^{2}.

$$8x^{2}-24x+16-6x-9=0$$

Pahekotia te -24x me -6x, ka -30x.

$$8x^{2}-30x+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$8x^{2}-30x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -30 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 8\times 7}}{2\times 8}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 8\times 7}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-30\right)±\sqrt{900-32\times 7}}{2\times 8}$$

Whakareatia -32 ki te 7.

$$x=\frac{-\left(-30\right)±\sqrt{900-224}}{2\times 8}$$

Tāpiri 900 ki te -224.

$$x=\frac{-\left(-30\right)±\sqrt{676}}{2\times 8}$$

Tuhia te pūtakerua o te 676.

$$x=\frac{-\left(-30\right)±26}{2\times 8}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±26}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{30±26}{16}$$

Nā, me whakaoti te whārite x=\frac{30±26}{16} ina he tāpiri te ±. Tāpiri 30 ki te 26.

$$x=\frac{56}{16}$$

Whakahekea te hautanga \frac{56}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=\frac{7}{2}$$

Nā, me whakaoti te whārite x=\frac{30±26}{16} ina he tango te ±. Tango 26 mai i 30.

$$x=\frac{4}{16}$$

Whakahekea te hautanga \frac{4}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{7}{2} x=\frac{1}{4}$$

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(3x-4\right)^{2}.

$$9x^{2}-24x+16-\left(x+3\right)^{2}=0$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+3\right)^{2}.

$$9x^{2}-24x+16-\left(x^{2}+6x+9\right)=0$$

Hei kimi i te tauaro o x^{2}+6x+9, kimihia te tauaro o ia taurangi.

$$9x^{2}-24x+16-x^{2}-6x-9=0$$

Pahekotia te 9x^{2} me -x^{2}, ka 8x^{2}.

$$8x^{2}-24x+16-6x-9=0$$

Pahekotia te -24x me -6x, ka -30x.

$$8x^{2}-30x+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$8x^{2}-30x+7=0$$

Tangohia te 7 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$8x^{2}-30x=-7$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-30x}{8}=-\frac{7}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\left(-\frac{30}{8}\right)x=-\frac{7}{8}$$

Whakahekea te hautanga \frac{-30}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{15}{4}x=-\frac{7}{8}$$

Whakawehea te -\frac{15}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{8}. Nā, tāpiria te pūrua o te -\frac{15}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{15}{4}x+\left(-\frac{15}{8}\right)^{2}=-\frac{7}{8}+\left(-\frac{15}{8}\right)^{2}$$

Pūruatia -\frac{15}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=-\frac{7}{8}+\frac{225}{64}$$

Tāpiri -\frac{7}{8} ki te \frac{225}{64} 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.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{169}{64}$$

Tauwehea x^{2}-\frac{15}{4}x+\frac{225}{64}. 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}.

$$\left(x-\frac{15}{8}\right)^{2}=\frac{169}{64}$$

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

$$\sqrt{\left(x-\frac{15}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Whakarūnātia.

$$x-\frac{15}{8}=\frac{13}{8} x-\frac{15}{8}=-\frac{13}{8}$$

Me tāpiri \frac{15}{8} ki ngā taha e rua o te whārite.

$$x=\frac{7}{2} x=\frac{1}{4}$$