Whakaoti i te 4(x+1)^2-169=0

$$4 ( x + 1 ) ^ { 2 } - 169 = 0$$

Whakaoti mō x

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

$$x = -\frac{15}{2} = -7\frac{1}{2} = -7.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$4\left(x^{2}+2x+1\right)-169=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}+2x+1.

$$4x^{2}+8x+4-169=0$$

Tangohia te 169 i te 4, ka -165.

$$4x^{2}+8x-165=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx-165. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=8 ab=4\left(-165\right)=-660$$

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 -660.

$$-1,660 -2,330 -3,220 -4,165 -5,132 -6,110 -10,66 -11,60 -12,55 -15,44 -20,33 -22,30$$

Tātaihia te tapeke mō ia takirua.

$$-1+660=659 -2+330=328 -3+220=217 -4+165=161 -5+132=127 -6+110=104 -10+66=56 -11+60=49 -12+55=43 -15+44=29 -20+33=13 -22+30=8$$

Ko te otinga te takirua ka hoatu i te tapeke 8.

$$a=-22 b=30$$

Tuhia anō te 4x^{2}+8x-165 hei \left(4x^{2}-22x\right)+\left(30x-165\right).

$$\left(4x^{2}-22x\right)+\left(30x-165\right)$$

Tauwehea te 2x i te tuatahi me te 15 i te rōpū tuarua.

$$2x\left(2x-11\right)+15\left(2x-11\right)$$

Whakatauwehea atu te kīanga pātahi 2x-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-11\right)\left(2x+15\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-11=0 me te 2x+15=0.

$$x=\frac{11}{2} x=-\frac{15}{2}$$

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(x+1\right)^{2}.

$$4\left(x^{2}+2x+1\right)-169=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}+2x+1.

$$4x^{2}+8x+4-169=0$$

Tangohia te 169 i te 4, ka -165.

$$4x^{2}+8x-165=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 8 mō b, me -165 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 4\left(-165\right)}}{2\times 4}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 4\left(-165\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-8±\sqrt{64-16\left(-165\right)}}{2\times 4}$$

Whakareatia -16 ki te -165.

$$x=\frac{-8±\sqrt{64+2640}}{2\times 4}$$

Tāpiri 64 ki te 2640.

$$x=\frac{-8±\sqrt{2704}}{2\times 4}$$

Tuhia te pūtakerua o te 2704.

$$x=\frac{-8±52}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-8±52}{8}$$

Nā, me whakaoti te whārite x=\frac{-8±52}{8} ina he tāpiri te ±. Tāpiri -8 ki te 52.

$$x=\frac{44}{8}$$

Whakahekea te hautanga \frac{44}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{11}{2}$$

Nā, me whakaoti te whārite x=\frac{-8±52}{8} ina he tango te ±. Tango 52 mai i -8.

$$x=-\frac{60}{8}$$

Whakahekea te hautanga \frac{-60}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{15}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{11}{2} x=-\frac{15}{2}$$

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(x+1\right)^{2}.

$$4\left(x^{2}+2x+1\right)-169=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x^{2}+2x+1.

$$4x^{2}+8x+4-169=0$$

Tangohia te 169 i te 4, ka -165.

$$4x^{2}+8x-165=0$$

Me tāpiri te 165 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$4x^{2}+8x=165$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+8x}{4}=\frac{165}{4}$$

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

$$x^{2}+\frac{8}{4}x=\frac{165}{4}$$

Whakawehe 8 ki te 4.

$$x^{2}+2x=\frac{165}{4}$$

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+2x+1^{2}=\frac{165}{4}+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=\frac{165}{4}+1$$

Tāpiri \frac{165}{4} ki te 1.

$$x^{2}+2x+1=\frac{169}{4}$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$x+1=\frac{13}{2} x+1=-\frac{13}{2}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=\frac{11}{2} x=-\frac{15}{2}$$