Whakaoti i te (7-a)(a-3)=1

$$( 7 - a ) ( a - 3 ) = 1$$

Whakaoti mō a

$$a=\sqrt{3}+5\approx 6.732050808$$

$$a=5-\sqrt{3}\approx 3.267949192$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 7-a ki te a-3 ka whakakotahi i ngā kupu rite.

$$10a-21-a^{2}=1$$

Tangohia te 1 mai i ngā taha e rua.

$$10a-21-a^{2}-1=0$$

Tangohia te 1 i te -21, ka -22.

$$10a-22-a^{2}=0$$

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.

$$-a^{2}+10a-22=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 10 mō b, me -22 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-22\right)}}{2\left(-1\right)}$$

Pūrua 10.

$$a=\frac{-10±\sqrt{100-4\left(-1\right)\left(-22\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$a=\frac{-10±\sqrt{100+4\left(-22\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -22.

$$a=\frac{-10±\sqrt{100-88}}{2\left(-1\right)}$$

Tāpiri 100 ki te -88.

$$a=\frac{-10±\sqrt{12}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 12.

$$a=\frac{-10±2\sqrt{3}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$a=\frac{-10±2\sqrt{3}}{-2}$$

Nā, me whakaoti te whārite a=\frac{-10±2\sqrt{3}}{-2} ina he tāpiri te ±. Tāpiri -10 ki te 2\sqrt{3}.

$$a=\frac{2\sqrt{3}-10}{-2}$$

Whakawehe -10+2\sqrt{3} ki te -2.

$$a=5-\sqrt{3}$$

Nā, me whakaoti te whārite a=\frac{-10±2\sqrt{3}}{-2} ina he tango te ±. Tango 2\sqrt{3} mai i -10.

$$a=\frac{-2\sqrt{3}-10}{-2}$$

Whakawehe -10-2\sqrt{3} ki te -2.

$$a=\sqrt{3}+5$$

Kua oti te whārite te whakatau.

$$a=5-\sqrt{3} a=\sqrt{3}+5$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 7-a ki te a-3 ka whakakotahi i ngā kupu rite.

$$10a-21-a^{2}=1$$

Me tāpiri te 21 ki ngā taha e rua.

$$10a-a^{2}=1+21$$

Tāpirihia te 1 ki te 21, ka 22.

$$10a-a^{2}=22$$

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.

$$-a^{2}+10a=22$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-a^{2}+10a}{-1}=\frac{22}{-1}$$

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

$$a^{2}+\frac{10}{-1}a=\frac{22}{-1}$$

Whakawehe 10 ki te -1.

$$a^{2}-10a=\frac{22}{-1}$$

Whakawehe 22 ki te -1.

$$a^{2}-10a=-22$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}-10a+\left(-5\right)^{2}=-22+\left(-5\right)^{2}$$

Pūrua -5.

$$a^{2}-10a+25=-22+25$$

Tāpiri -22 ki te 25.

$$a^{2}-10a+25=3$$

Tauwehea a^{2}-10a+25. 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}.

$$\left(a-5\right)^{2}=3$$

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

$$\sqrt{\left(a-5\right)^{2}}=\sqrt{3}$$

Whakarūnātia.

$$a-5=\sqrt{3} a-5=-\sqrt{3}$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$a=\sqrt{3}+5 a=5-\sqrt{3}$$