Whakaoti i te 5a^2-a-5a+1=12a^2-5a-6a

$$5 a ^ { 2 } - a - 5 a + 1 = 12 a ^ { 2 } - 5 a - 6 a$$

Whakaoti mō a

$$a=\frac{\sqrt{53}+5}{14}\approx 0.877150706$$

$$a=\frac{5-\sqrt{53}}{14}\approx -0.162864992$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te -a me -5a, ka -6a.

$$5a^{2}-6a+1=12a^{2}-5a-6a$$

Pahekotia te -5a me -6a, ka -11a.

$$5a^{2}-6a+1=12a^{2}-11a$$

Tangohia te 12a^{2} mai i ngā taha e rua.

$$5a^{2}-6a+1-12a^{2}=-11a$$

Pahekotia te 5a^{2} me -12a^{2}, ka -7a^{2}.

$$-7a^{2}-6a+1=-11a$$

Me tāpiri te 11a ki ngā taha e rua.

$$-7a^{2}-6a+1+11a=0$$

Pahekotia te -6a me 11a, ka 5a.

$$-7a^{2}+5a+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -7 mō a, 5 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 5.

$$a=\frac{-5±\sqrt{25-4\left(-7\right)}}{2\left(-7\right)}$$

Whakareatia -4 ki te -7.

$$a=\frac{-5±\sqrt{25+28}}{2\left(-7\right)}$$

Tāpiri 25 ki te 28.

$$a=\frac{-5±\sqrt{53}}{2\left(-7\right)}$$

Whakareatia 2 ki te -7.

$$a=\frac{-5±\sqrt{53}}{-14}$$

Nā, me whakaoti te whārite a=\frac{-5±\sqrt{53}}{-14} ina he tāpiri te ±. Tāpiri -5 ki te \sqrt{53}.

$$a=\frac{\sqrt{53}-5}{-14}$$

Whakawehe -5+\sqrt{53} ki te -14.

$$a=\frac{5-\sqrt{53}}{14}$$

Nā, me whakaoti te whārite a=\frac{-5±\sqrt{53}}{-14} ina he tango te ±. Tango \sqrt{53} mai i -5.

$$a=\frac{-\sqrt{53}-5}{-14}$$

Whakawehe -5-\sqrt{53} ki te -14.

$$a=\frac{\sqrt{53}+5}{14}$$

Kua oti te whārite te whakatau.

$$a=\frac{5-\sqrt{53}}{14} a=\frac{\sqrt{53}+5}{14}$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te -a me -5a, ka -6a.

$$5a^{2}-6a+1=12a^{2}-5a-6a$$

Pahekotia te -5a me -6a, ka -11a.

$$5a^{2}-6a+1=12a^{2}-11a$$

Tangohia te 12a^{2} mai i ngā taha e rua.

$$5a^{2}-6a+1-12a^{2}=-11a$$

Pahekotia te 5a^{2} me -12a^{2}, ka -7a^{2}.

$$-7a^{2}-6a+1=-11a$$

Me tāpiri te 11a ki ngā taha e rua.

$$-7a^{2}-6a+1+11a=0$$

Pahekotia te -6a me 11a, ka 5a.

$$-7a^{2}+5a+1=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-7a^{2}+5a=-1$$

Whakawehea ngā taha e rua ki te -7.

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

Mā te whakawehe ki te -7 ka wetekia te whakareanga ki te -7.

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

Whakawehe 5 ki te -7.

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

Whakawehe -1 ki te -7.

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

Whakawehea te -\frac{5}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{14}. Nā, tāpiria te pūrua o te -\frac{5}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}-\frac{5}{7}a+\left(-\frac{5}{14}\right)^{2}=\frac{1}{7}+\left(-\frac{5}{14}\right)^{2}$$

Pūruatia -\frac{5}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$a^{2}-\frac{5}{7}a+\frac{25}{196}=\frac{1}{7}+\frac{25}{196}$$

Tāpiri \frac{1}{7} ki te \frac{25}{196} 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.

$$a^{2}-\frac{5}{7}a+\frac{25}{196}=\frac{53}{196}$$

Tauwehea te a^{2}-\frac{5}{7}a+\frac{25}{196}. 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}.

$$\left(a-\frac{5}{14}\right)^{2}=\frac{53}{196}$$

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

$$\sqrt{\left(a-\frac{5}{14}\right)^{2}}=\sqrt{\frac{53}{196}}$$

Whakarūnātia.

$$a-\frac{5}{14}=\frac{\sqrt{53}}{14} a-\frac{5}{14}=-\frac{\sqrt{53}}{14}$$

Me tāpiri \frac{5}{14} ki ngā taha e rua o te whārite.

$$a=\frac{\sqrt{53}+5}{14} a=\frac{5-\sqrt{53}}{14}$$