Whakaoti i te left(x+14right)^2-left(x+11right)^2=left(x-6right)^2

$${ \left(x+14 \right) }^{ 2 } - { \left(x+11 \right) }^{ 2 } = { \left(x-6 \right) }^{ 2 }$$

Whakaoti mō x

$$x=2\sqrt{30}+9\approx 19.95445115$$

$$x=9-2\sqrt{30}\approx -1.95445115$$

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(x+14\right)^{2}.

$$x^{2}+28x+196-\left(x+11\right)^{2}=\left(x-6\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+11\right)^{2}.

$$x^{2}+28x+196-\left(x^{2}+22x+121\right)=\left(x-6\right)^{2}$$

Hei kimi i te tauaro o x^{2}+22x+121, kimihia te tauaro o ia taurangi.

$$x^{2}+28x+196-x^{2}-22x-121=\left(x-6\right)^{2}$$

Pahekotia te x^{2} me -x^{2}, ka 0.

$$28x+196-22x-121=\left(x-6\right)^{2}$$

Pahekotia te 28x me -22x, ka 6x.

$$6x+196-121=\left(x-6\right)^{2}$$

Tangohia te 121 i te 196, ka 75.

$$6x+75=\left(x-6\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-6\right)^{2}.

$$6x+75=x^{2}-12x+36$$

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

$$6x+75-x^{2}=-12x+36$$

Me tāpiri te 12x ki ngā taha e rua.

$$6x+75-x^{2}+12x=36$$

Pahekotia te 6x me 12x, ka 18x.

$$18x+75-x^{2}=36$$

Tangohia te 36 mai i ngā taha e rua.

$$18x+75-x^{2}-36=0$$

Tangohia te 36 i te 75, ka 39.

$$18x+39-x^{2}=0$$

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.

$$-x^{2}+18x+39=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 18 mō b, me 39 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-1\right)\times 39}}{2\left(-1\right)}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\left(-1\right)\times 39}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-18±\sqrt{324+4\times 39}}{2\left(-1\right)}$$

Whakareatia 4 ki te 39.

$$x=\frac{-18±\sqrt{324+156}}{2\left(-1\right)}$$

Tāpiri 324 ki te 156.

$$x=\frac{-18±\sqrt{480}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 480.

$$x=\frac{-18±4\sqrt{30}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-18±4\sqrt{30}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-18±4\sqrt{30}}{-2} ina he tāpiri te ±. Tāpiri -18 ki te 4\sqrt{30}.

$$x=\frac{4\sqrt{30}-18}{-2}$$

Whakawehe -18+4\sqrt{30} ki te -2.

$$x=9-2\sqrt{30}$$

Nā, me whakaoti te whārite x=\frac{-18±4\sqrt{30}}{-2} ina he tango te ±. Tango 4\sqrt{30} mai i -18.

$$x=\frac{-4\sqrt{30}-18}{-2}$$

Whakawehe -18-4\sqrt{30} ki te -2.

$$x=2\sqrt{30}+9$$

Kua oti te whārite te whakatau.

$$x=9-2\sqrt{30} x=2\sqrt{30}+9$$

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(x+14\right)^{2}.

$$x^{2}+28x+196-\left(x+11\right)^{2}=\left(x-6\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+11\right)^{2}.

$$x^{2}+28x+196-\left(x^{2}+22x+121\right)=\left(x-6\right)^{2}$$

Hei kimi i te tauaro o x^{2}+22x+121, kimihia te tauaro o ia taurangi.

$$x^{2}+28x+196-x^{2}-22x-121=\left(x-6\right)^{2}$$

Pahekotia te x^{2} me -x^{2}, ka 0.

$$28x+196-22x-121=\left(x-6\right)^{2}$$

Pahekotia te 28x me -22x, ka 6x.

$$6x+196-121=\left(x-6\right)^{2}$$

Tangohia te 121 i te 196, ka 75.

$$6x+75=\left(x-6\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-6\right)^{2}.

$$6x+75=x^{2}-12x+36$$

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

$$6x+75-x^{2}=-12x+36$$

Me tāpiri te 12x ki ngā taha e rua.

$$6x+75-x^{2}+12x=36$$

Pahekotia te 6x me 12x, ka 18x.

$$18x+75-x^{2}=36$$

Tangohia te 75 mai i ngā taha e rua.

$$18x-x^{2}=36-75$$

Tangohia te 75 i te 36, ka -39.

$$18x-x^{2}=-39$$

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.

$$-x^{2}+18x=-39$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+18x}{-1}=-\frac{39}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{18}{-1}x=-\frac{39}{-1}$$

Whakawehe 18 ki te -1.

$$x^{2}-18x=-\frac{39}{-1}$$

Whakawehe -39 ki te -1.

$$x^{2}-18x=39$$

Whakawehea te -18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -9. Nā, tāpiria te pūrua o te -9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-18x+\left(-9\right)^{2}=39+\left(-9\right)^{2}$$

Pūrua -9.

$$x^{2}-18x+81=39+81$$

Tāpiri 39 ki te 81.

$$x^{2}-18x+81=120$$

Tauwehea te x^{2}-18x+81. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=120$$

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

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{120}$$

Whakarūnātia.

$$x-9=2\sqrt{30} x-9=-2\sqrt{30}$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x=2\sqrt{30}+9 x=9-2\sqrt{30}$$