Whakaoti i te -1/2x+1=-1/4x^2+3/2x+4

$$- \frac { 1 } { 2 } x + 1 = - \frac { 1 } { 4 } x ^ { 2 } + \frac { 3 } { 2 } x + 4$$

Whakaoti mō x

$$x=2\sqrt{7}+4\approx 9.291502622$$

$$x=4-2\sqrt{7}\approx -1.291502622$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te \frac{1}{4}x^{2} ki ngā taha e rua.

$$-\frac{1}{2}x+1+\frac{1}{4}x^{2}=\frac{3}{2}x+4$$

Tangohia te \frac{3}{2}x mai i ngā taha e rua.

$$-\frac{1}{2}x+1+\frac{1}{4}x^{2}-\frac{3}{2}x=4$$

Pahekotia te -\frac{1}{2}x me -\frac{3}{2}x, ka -2x.

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

Tangohia te 4 mai i ngā taha e rua.

$$-2x+1+\frac{1}{4}x^{2}-4=0$$

Tangohia te 4 i te 1, ka -3.

$$-2x-3+\frac{1}{4}x^{2}=0$$

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.

$$\frac{1}{4}x^{2}-2x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{4} mō a, -2 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times \frac{1}{4}\left(-3\right)}}{2\times \frac{1}{4}}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times \frac{1}{4}\left(-3\right)}}{2\times \frac{1}{4}}$$

Whakareatia -4 ki te \frac{1}{4}.

$$x=\frac{-\left(-2\right)±\sqrt{4-\left(-3\right)}}{2\times \frac{1}{4}}$$

Whakareatia -1 ki te -3.

$$x=\frac{-\left(-2\right)±\sqrt{4+3}}{2\times \frac{1}{4}}$$

Tāpiri 4 ki te 3.

$$x=\frac{-\left(-2\right)±\sqrt{7}}{2\times \frac{1}{4}}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±\sqrt{7}}{2\times \frac{1}{4}}$$

Whakareatia 2 ki te \frac{1}{4}.

$$x=\frac{2±\sqrt{7}}{\frac{1}{2}}$$

Nā, me whakaoti te whārite x=\frac{2±\sqrt{7}}{\frac{1}{2}} ina he tāpiri te ±. Tāpiri 2 ki te \sqrt{7}.

$$x=\frac{\sqrt{7}+2}{\frac{1}{2}}$$

Whakawehe 2+\sqrt{7} ki te \frac{1}{2} mā te whakarea 2+\sqrt{7} ki te tau huripoki o \frac{1}{2}.

$$x=2\sqrt{7}+4$$

Nā, me whakaoti te whārite x=\frac{2±\sqrt{7}}{\frac{1}{2}} ina he tango te ±. Tango \sqrt{7} mai i 2.

$$x=\frac{2-\sqrt{7}}{\frac{1}{2}}$$

Whakawehe 2-\sqrt{7} ki te \frac{1}{2} mā te whakarea 2-\sqrt{7} ki te tau huripoki o \frac{1}{2}.

$$x=4-2\sqrt{7}$$

Kua oti te whārite te whakatau.

$$x=2\sqrt{7}+4 x=4-2\sqrt{7}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te \frac{1}{4}x^{2} ki ngā taha e rua.

$$-\frac{1}{2}x+1+\frac{1}{4}x^{2}=\frac{3}{2}x+4$$

Tangohia te \frac{3}{2}x mai i ngā taha e rua.

$$-\frac{1}{2}x+1+\frac{1}{4}x^{2}-\frac{3}{2}x=4$$

Pahekotia te -\frac{1}{2}x me -\frac{3}{2}x, ka -2x.

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

Tangohia te 1 mai i ngā taha e rua.

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

Tangohia te 1 i te 4, ka 3.

$$-2x+\frac{1}{4}x^{2}=3$$

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.

$$\frac{1}{4}x^{2}-2x=3$$

Me whakarea ngā taha e rua ki te 4.

$$\frac{\frac{1}{4}x^{2}-2x}{\frac{1}{4}}=\frac{3}{\frac{1}{4}}$$

Mā te whakawehe ki te \frac{1}{4} ka wetekia te whakareanga ki te \frac{1}{4}.

$$x^{2}+\left(-\frac{2}{\frac{1}{4}}\right)x=\frac{3}{\frac{1}{4}}$$

Whakawehe -2 ki te \frac{1}{4} mā te whakarea -2 ki te tau huripoki o \frac{1}{4}.

$$x^{2}-8x=\frac{3}{\frac{1}{4}}$$

Whakawehe 3 ki te \frac{1}{4} mā te whakarea 3 ki te tau huripoki o \frac{1}{4}.

$$x^{2}-8x=12$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=12+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=12+16$$

Tāpiri 12 ki te 16.

$$x^{2}-8x+16=28$$

Tauwehea x^{2}-8x+16. 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}.

$$\left(x-4\right)^{2}=28$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{28}$$

Whakarūnātia.

$$x-4=2\sqrt{7} x-4=-2\sqrt{7}$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=2\sqrt{7}+4 x=4-2\sqrt{7}$$