Whakaoti i te (3x-1)^2-(2x+1)^2=7

$$( 3 x - 1 ) ^ { 2 } - ( 2 x + 1 ) ^ { 2 } = 7$$

Whakaoti mō x

$$x=\frac{2\sqrt{15}}{5}+1\approx 2.549193338$$

$$x=-\frac{2\sqrt{15}}{5}+1\approx -0.549193338$$

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-1\right)^{2}.

$$9x^{2}-6x+1-\left(2x+1\right)^{2}=7$$

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

$$9x^{2}-6x+1-\left(4x^{2}+4x+1\right)=7$$

Hei kimi i te tauaro o 4x^{2}+4x+1, kimihia te tauaro o ia taurangi.

$$9x^{2}-6x+1-4x^{2}-4x-1=7$$

Pahekotia te 9x^{2} me -4x^{2}, ka 5x^{2}.

$$5x^{2}-6x+1-4x-1=7$$

Pahekotia te -6x me -4x, ka -10x.

$$5x^{2}-10x+1-1=7$$

Tangohia te 1 i te 1, ka 0.

$$5x^{2}-10x=7$$

Tangohia te 7 mai i ngā taha e rua.

$$5x^{2}-10x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -10 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 5\left(-7\right)}}{2\times 5}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 5\left(-7\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-10\right)±\sqrt{100-20\left(-7\right)}}{2\times 5}$$

Whakareatia -20 ki te -7.

$$x=\frac{-\left(-10\right)±\sqrt{100+140}}{2\times 5}$$

Tāpiri 100 ki te 140.

$$x=\frac{-\left(-10\right)±\sqrt{240}}{2\times 5}$$

Tuhia te pūtakerua o te 240.

$$x=\frac{-\left(-10\right)±4\sqrt{15}}{2\times 5}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±4\sqrt{15}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{10±4\sqrt{15}}{10}$$

Nā, me whakaoti te whārite x=\frac{10±4\sqrt{15}}{10} ina he tāpiri te ±. Tāpiri 10 ki te 4\sqrt{15}.

$$x=\frac{4\sqrt{15}+10}{10}$$

Whakawehe 10+4\sqrt{15} ki te 10.

$$x=\frac{2\sqrt{15}}{5}+1$$

Nā, me whakaoti te whārite x=\frac{10±4\sqrt{15}}{10} ina he tango te ±. Tango 4\sqrt{15} mai i 10.

$$x=\frac{10-4\sqrt{15}}{10}$$

Whakawehe 10-4\sqrt{15} ki te 10.

$$x=-\frac{2\sqrt{15}}{5}+1$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{15}}{5}+1 x=-\frac{2\sqrt{15}}{5}+1$$

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-1\right)^{2}.

$$9x^{2}-6x+1-\left(2x+1\right)^{2}=7$$

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

$$9x^{2}-6x+1-\left(4x^{2}+4x+1\right)=7$$

Hei kimi i te tauaro o 4x^{2}+4x+1, kimihia te tauaro o ia taurangi.

$$9x^{2}-6x+1-4x^{2}-4x-1=7$$

Pahekotia te 9x^{2} me -4x^{2}, ka 5x^{2}.

$$5x^{2}-6x+1-4x-1=7$$

Pahekotia te -6x me -4x, ka -10x.

$$5x^{2}-10x+1-1=7$$

Tangohia te 1 i te 1, ka 0.

$$5x^{2}-10x=7$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-10x}{5}=\frac{7}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\left(-\frac{10}{5}\right)x=\frac{7}{5}$$

Whakawehe -10 ki te 5.

$$x^{2}-2x=\frac{7}{5}$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=\frac{7}{5}+1$$

Tāpiri \frac{7}{5} ki te 1.

$$x^{2}-2x+1=\frac{12}{5}$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=\frac{12}{5}$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{12}{5}}$$

Whakarūnātia.

$$x-1=\frac{2\sqrt{15}}{5} x-1=-\frac{2\sqrt{15}}{5}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{15}}{5}+1 x=-\frac{2\sqrt{15}}{5}+1$$