Whakaoti i te 4x-1/2x^2/2=15/4

$$\frac{ 4x- \frac{ 1 }{ 2 } { x }^{ 2 } }{ 2 } = \frac{ 15 }{ 4 }$$

Whakaoti mō x

$$x=3$$

$$x=5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua ki te 2.

$$4x-\frac{1}{2}x^{2}=\frac{15}{4}\times 2$$

Whakareatia te \frac{15}{4} ki te 2, ka \frac{15}{2}.

$$4x-\frac{1}{2}x^{2}=\frac{15}{2}$$

Tangohia te \frac{15}{2} mai i ngā taha e rua.

$$4x-\frac{1}{2}x^{2}-\frac{15}{2}=0$$

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.

$$-\frac{1}{2}x^{2}+4x-\frac{15}{2}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{2} mō a, 4 mō b, me -\frac{15}{2} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-\frac{1}{2}\right)\left(-\frac{15}{2}\right)}}{2\left(-\frac{1}{2}\right)}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-\frac{1}{2}\right)\left(-\frac{15}{2}\right)}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-4±\sqrt{16+2\left(-\frac{15}{2}\right)}}{2\left(-\frac{1}{2}\right)}$$

Whakareatia 2 ki te -\frac{15}{2}.

$$x=\frac{-4±\sqrt{16-15}}{2\left(-\frac{1}{2}\right)}$$

Tāpiri 16 ki te -15.

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

Tuhia te pūtakerua o te 1.

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

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

$$x=\frac{-4±1}{-1}$$

Nā, me whakaoti te whārite x=\frac{-4±1}{-1} ina he tāpiri te ±. Tāpiri -4 ki te 1.

$$x=-\frac{3}{-1}$$

Whakawehe -3 ki te -1.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{-4±1}{-1} ina he tango te ±. Tango 1 mai i -4.

$$x=-\frac{5}{-1}$$

Whakawehe -5 ki te -1.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=3 x=5$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua ki te 2.

$$4x-\frac{1}{2}x^{2}=\frac{15}{4}\times 2$$

Whakareatia te \frac{15}{4} ki te 2, ka \frac{15}{2}.

$$4x-\frac{1}{2}x^{2}=\frac{15}{2}$$

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.

$$-\frac{1}{2}x^{2}+4x=\frac{15}{2}$$

Me whakarea ngā taha e rua ki te -2.

$$\frac{-\frac{1}{2}x^{2}+4x}{-\frac{1}{2}}=\frac{\frac{15}{2}}{-\frac{1}{2}}$$

Mā te whakawehe ki te -\frac{1}{2} ka wetekia te whakareanga ki te -\frac{1}{2}.

$$x^{2}+\frac{4}{-\frac{1}{2}}x=\frac{\frac{15}{2}}{-\frac{1}{2}}$$

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

$$x^{2}-8x=\frac{\frac{15}{2}}{-\frac{1}{2}}$$

Whakawehe \frac{15}{2} ki te -\frac{1}{2} mā te whakarea \frac{15}{2} ki te tau huripoki o -\frac{1}{2}.

$$x^{2}-8x=-15$$

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.

$$x^{2}-8x+\left(-4\right)^{2}=-15+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-15+16$$

Tāpiri -15 ki te 16.

$$x^{2}-8x+16=1$$

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}.

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

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

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

Whakarūnātia.

$$x-4=1 x-4=-1$$

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

$$x=5 x=3$$