Whakaoti i te (-7x-2)^2-25=0

$$( - 7 x - 2 ) ^ { 2 } - 25 = 0$$

Whakaoti mō x

$$x=-1$$

$$x=\frac{3}{7}\approx 0.428571429$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-7x-2\right)^{2}.

$$49x^{2}+28x+4-25=0$$

Tangohia te 25 i te 4, ka -21.

$$49x^{2}+28x-21=0$$

Whakawehea ngā taha e rua ki te 7.

$$7x^{2}+4x-3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 7x^{2}+ax+bx-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=7\left(-3\right)=-21$$

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 -21.

$$-1,21 -3,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+21=20 -3+7=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-3 b=7$$

Tuhia anō te 7x^{2}+4x-3 hei \left(7x^{2}-3x\right)+\left(7x-3\right).

$$\left(7x^{2}-3x\right)+\left(7x-3\right)$$

Whakatauwehea atu x i te 7x^{2}-3x.

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

Whakatauwehea atu te kīanga pātahi 7x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-3\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 7x-3=0 me te x+1=0.

$$x=\frac{3}{7} x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-7x-2\right)^{2}.

$$49x^{2}+28x+4-25=0$$

Tangohia te 25 i te 4, ka -21.

$$49x^{2}+28x-21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 49 mō a, 28 mō b, me -21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 49\left(-21\right)}}{2\times 49}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\times 49\left(-21\right)}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$x=\frac{-28±\sqrt{784-196\left(-21\right)}}{2\times 49}$$

Whakareatia -196 ki te -21.

$$x=\frac{-28±\sqrt{784+4116}}{2\times 49}$$

Tāpiri 784 ki te 4116.

$$x=\frac{-28±\sqrt{4900}}{2\times 49}$$

Tuhia te pūtakerua o te 4900.

$$x=\frac{-28±70}{2\times 49}$$

Whakareatia 2 ki te 49.

$$x=\frac{-28±70}{98}$$

Nā, me whakaoti te whārite x=\frac{-28±70}{98} ina he tāpiri te ±. Tāpiri -28 ki te 70.

$$x=\frac{42}{98}$$

Whakahekea te hautanga \frac{42}{98} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$x=\frac{3}{7}$$

Nā, me whakaoti te whārite x=\frac{-28±70}{98} ina he tango te ±. Tango 70 mai i -28.

$$x=-\frac{98}{98}$$

Whakawehe -98 ki te 98.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=\frac{3}{7} x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-7x-2\right)^{2}.

$$49x^{2}+28x+4-25=0$$

Tangohia te 25 i te 4, ka -21.

$$49x^{2}+28x-21=0$$

Me tāpiri te 21 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$49x^{2}+28x=21$$

Whakawehea ngā taha e rua ki te 49.

$$\frac{49x^{2}+28x}{49}=\frac{21}{49}$$

Mā te whakawehe ki te 49 ka wetekia te whakareanga ki te 49.

$$x^{2}+\frac{28}{49}x=\frac{21}{49}$$

Whakahekea te hautanga \frac{28}{49} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 7.

$$x^{2}+\frac{4}{7}x=\frac{21}{49}$$

Whakahekea te hautanga \frac{21}{49} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 7.

$$x^{2}+\frac{4}{7}x=\frac{3}{7}$$

Whakawehea te \frac{4}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{2}{7}. Nā, tāpiria te pūrua o te \frac{2}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{4}{7}x+\left(\frac{2}{7}\right)^{2}=\frac{3}{7}+\left(\frac{2}{7}\right)^{2}$$

Pūruatia \frac{2}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=\frac{3}{7}+\frac{4}{49}$$

Tāpiri \frac{3}{7} ki te \frac{4}{49} 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.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=\frac{25}{49}$$

Tauwehea x^{2}+\frac{4}{7}x+\frac{4}{49}. 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}.

$$\left(x+\frac{2}{7}\right)^{2}=\frac{25}{49}$$

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

$$\sqrt{\left(x+\frac{2}{7}\right)^{2}}=\sqrt{\frac{25}{49}}$$

Whakarūnātia.

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

Me tango \frac{2}{7} mai i ngā taha e rua o te whārite.

$$x=\frac{3}{7} x=-1$$