Whakaoti i te 4x^2+20x+25=49

$$4 x ^ { 2 } + 20 x + 25 = 49$$

Whakaoti mō x

$$x=-6$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 49 mai i ngā taha e rua.

$$4x^{2}+20x+25-49=0$$

Tangohia te 49 i te 25, ka -24.

$$4x^{2}+20x-24=0$$

Whakawehea ngā taha e rua ki te 4.

$$x^{2}+5x-6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=1\left(-6\right)=-6$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -6.

$$-1,6 -2,3$$

Tātaihia te tapeke mō ia takirua.

$$-1+6=5 -2+3=1$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-1 b=6$$

Tuhia anō te x^{2}+5x-6 hei \left(x^{2}-x\right)+\left(6x-6\right).

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

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-1\right)+6\left(x-1\right)$$

Whakatauwehea atu te kīanga pātahi x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-1\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te x+6=0.

$$x=1 x=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$4x^{2}+20x+25=49$$

Me tango 49 mai i ngā taha e rua o te whārite.

$$4x^{2}+20x+25-49=49-49$$

Mā te tango i te 49 i a ia ake anō ka toe ko te 0.

$$4x^{2}+20x+25-49=0$$

Tango 49 mai i 25.

$$4x^{2}+20x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 20 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 4\left(-24\right)}}{2\times 4}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\times 4\left(-24\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-20±\sqrt{400-16\left(-24\right)}}{2\times 4}$$

Whakareatia -16 ki te -24.

$$x=\frac{-20±\sqrt{400+384}}{2\times 4}$$

Tāpiri 400 ki te 384.

$$x=\frac{-20±\sqrt{784}}{2\times 4}$$

Tuhia te pūtakerua o te 784.

$$x=\frac{-20±28}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-20±28}{8}$$

Nā, me whakaoti te whārite x=\frac{-20±28}{8} ina he tāpiri te ±. Tāpiri -20 ki te 28.

$$x=\frac{8}{8}$$

Whakawehe 8 ki te 8.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-20±28}{8} ina he tango te ±. Tango 28 mai i -20.

$$x=-\frac{48}{8}$$

Whakawehe -48 ki te 8.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=1 x=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$4x^{2}+20x+25=49$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$4x^{2}+20x+25-25=49-25$$

Mā te tango i te 25 i a ia ake anō ka toe ko te 0.

$$4x^{2}+20x=49-25$$

Tango 25 mai i 49.

$$4x^{2}+20x=24$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+20x}{4}=\frac{24}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\frac{20}{4}x=\frac{24}{4}$$

Whakawehe 20 ki te 4.

$$x^{2}+5x=\frac{24}{4}$$

Whakawehe 24 ki te 4.

$$x^{2}+5x=6$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=6+\left(\frac{5}{2}\right)^{2}$$

Pūruatia \frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+5x+\frac{25}{4}=6+\frac{25}{4}$$

Tāpiri 6 ki te \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{49}{4}$$

Tauwehea x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x+\frac{5}{2}=\frac{7}{2} x+\frac{5}{2}=-\frac{7}{2}$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$x=1 x=-6$$