Whakaoti i te 0=-1/4x^2+3/2x+4

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

Whakaoti mō x

$$x=-2$$

$$x=8$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

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

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

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

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}-4\left(-\frac{1}{4}\right)\times 4}}{2\left(-\frac{1}{4}\right)}$$

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

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}+4}}{2\left(-\frac{1}{4}\right)}$$

Tāpiri \frac{9}{4} ki te 4.

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{25}{4}}}{2\left(-\frac{1}{4}\right)}$$

Tuhia te pūtakerua o te \frac{25}{4}.

$$x=\frac{-\frac{3}{2}±\frac{5}{2}}{2\left(-\frac{1}{4}\right)}$$

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

$$x=\frac{-\frac{3}{2}±\frac{5}{2}}{-\frac{1}{2}}$$

Nā, me whakaoti te whārite x=\frac{-\frac{3}{2}±\frac{5}{2}}{-\frac{1}{2}} ina he tāpiri te ±. Tāpiri -\frac{3}{2} ki te \frac{5}{2} 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.

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

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

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-\frac{3}{2}±\frac{5}{2}}{-\frac{1}{2}} ina he tango te ±. Tango \frac{5}{2} mai i -\frac{3}{2} mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

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

$$x=8$$

Kua oti te whārite te whakatau.

$$x=-2 x=8$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

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

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Me whakarea ngā taha e rua ki te -4.

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

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

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

Whakawehe \frac{3}{2} ki te -\frac{1}{4} mā te whakarea \frac{3}{2} ki te tau huripoki o -\frac{1}{4}.

$$x^{2}-6x=-\frac{4}{-\frac{1}{4}}$$

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

$$x^{2}-6x=16$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=16+9$$

Tāpiri 16 ki te 9.

$$x^{2}-6x+9=25$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=25$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-3=5 x-3=-5$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=8 x=-2$$