Whakaoti i te left(7+xright)left(3-xright)=5

$$\left( 7+x \right) \left( 3-x \right) =5$$

Whakaoti mō x

$$x=2\sqrt{5}-2\approx 2.472135955$$

$$x=-2\sqrt{5}-2\approx -6.472135955$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$21-4x-x^{2}=5$$

Tangohia te 5 mai i ngā taha e rua.

$$21-4x-x^{2}-5=0$$

Tangohia te 5 i te 21, ka 16.

$$16-4x-x^{2}=0$$

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.

$$-x^{2}-4x+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -4 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 16}}{2\left(-1\right)}$$

Whakareatia 4 ki te 16.

$$x=\frac{-\left(-4\right)±\sqrt{16+64}}{2\left(-1\right)}$$

Tāpiri 16 ki te 64.

$$x=\frac{-\left(-4\right)±\sqrt{80}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 80.

$$x=\frac{-\left(-4\right)±4\sqrt{5}}{2\left(-1\right)}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±4\sqrt{5}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{4±4\sqrt{5}}{-2}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{5}}{-2} ina he tāpiri te ±. Tāpiri 4 ki te 4\sqrt{5}.

$$x=\frac{4\sqrt{5}+4}{-2}$$

Whakawehe 4+4\sqrt{5} ki te -2.

$$x=-2\sqrt{5}-2$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{5}}{-2} ina he tango te ±. Tango 4\sqrt{5} mai i 4.

$$x=\frac{4-4\sqrt{5}}{-2}$$

Whakawehe 4-4\sqrt{5} ki te -2.

$$x=2\sqrt{5}-2$$

Kua oti te whārite te whakatau.

$$x=-2\sqrt{5}-2 x=2\sqrt{5}-2$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$21-4x-x^{2}=5$$

Tangohia te 21 mai i ngā taha e rua.

$$-4x-x^{2}=5-21$$

Tangohia te 21 i te 5, ka -16.

$$-4x-x^{2}=-16$$

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.

$$-x^{2}-4x=-16$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{16}{-1}$$

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

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{16}{-1}$$

Whakawehe -4 ki te -1.

$$x^{2}+4x=-\frac{16}{-1}$$

Whakawehe -16 ki te -1.

$$x^{2}+4x=16$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=16+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=16+4$$

Tāpiri 16 ki te 4.

$$x^{2}+4x+4=20$$

Tauwehea x^{2}+4x+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}.

$$\left(x+2\right)^{2}=20$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{20}$$

Whakarūnātia.

$$x+2=2\sqrt{5} x+2=-2\sqrt{5}$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=2\sqrt{5}-2 x=-2\sqrt{5}-2$$