Whakaoti i te left(12-xright)^2+144=9x^2

$${ \left(12-x \right) }^{ 2 } +144=9 { x }^{ 2 }$$

Whakaoti mō x

$$x = \frac{3 \sqrt{17} - 3}{2} \approx 4.684658438$$

$$x=\frac{-3\sqrt{17}-3}{2}\approx -7.684658438$$

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(12-x\right)^{2}.

$$144-24x+x^{2}+144=9x^{2}$$

Tāpirihia te 144 ki te 144, ka 288.

$$288-24x+x^{2}=9x^{2}$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$288-24x+x^{2}-9x^{2}=0$$

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

$$288-24x-8x^{2}=0$$

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.

$$-8x^{2}-24x+288=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, -24 mō b, me 288 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-8\right)\times 288}}{2\left(-8\right)}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-8\right)\times 288}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-\left(-24\right)±\sqrt{576+32\times 288}}{2\left(-8\right)}$$

Whakareatia 32 ki te 288.

$$x=\frac{-\left(-24\right)±\sqrt{576+9216}}{2\left(-8\right)}$$

Tāpiri 576 ki te 9216.

$$x=\frac{-\left(-24\right)±\sqrt{9792}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 9792.

$$x=\frac{-\left(-24\right)±24\sqrt{17}}{2\left(-8\right)}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±24\sqrt{17}}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{24±24\sqrt{17}}{-16}$$

Nā, me whakaoti te whārite x=\frac{24±24\sqrt{17}}{-16} ina he tāpiri te ±. Tāpiri 24 ki te 24\sqrt{17}.

$$x=\frac{24\sqrt{17}+24}{-16}$$

Whakawehe 24+24\sqrt{17} ki te -16.

$$x=\frac{-3\sqrt{17}-3}{2}$$

Nā, me whakaoti te whārite x=\frac{24±24\sqrt{17}}{-16} ina he tango te ±. Tango 24\sqrt{17} mai i 24.

$$x=\frac{24-24\sqrt{17}}{-16}$$

Whakawehe 24-24\sqrt{17} ki te -16.

$$x=\frac{3\sqrt{17}-3}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3\sqrt{17}-3}{2} x=\frac{3\sqrt{17}-3}{2}$$

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(12-x\right)^{2}.

$$144-24x+x^{2}+144=9x^{2}$$

Tāpirihia te 144 ki te 144, ka 288.

$$288-24x+x^{2}=9x^{2}$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$288-24x+x^{2}-9x^{2}=0$$

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

$$288-24x-8x^{2}=0$$

Tangohia te 288 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-24x-8x^{2}=-288$$

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.

$$-8x^{2}-24x=-288$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8x^{2}-24x}{-8}=-\frac{288}{-8}$$

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

$$x^{2}+\left(-\frac{24}{-8}\right)x=-\frac{288}{-8}$$

Whakawehe -24 ki te -8.

$$x^{2}+3x=-\frac{288}{-8}$$

Whakawehe -288 ki te -8.

$$x^{2}+3x=36$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=36+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=36+\frac{9}{4}$$

Tāpiri 36 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{153}{4}$$

Tauwehea x^{2}+3x+\frac{9}{4}. 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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{153}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{153}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{3\sqrt{17}}{2} x+\frac{3}{2}=-\frac{3\sqrt{17}}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{17}-3}{2} x=\frac{-3\sqrt{17}-3}{2}$$