Whakaoti i te (x+5)(2x+7)-(x+5)(x-3)=0

$$( x + 5 ) ( 2 x + 7 ) - ( x + 5 ) ( x - 3 ) = 0$$

Whakaoti mō x

$$x=-10$$

$$x=-5$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te 2x+7 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x+5\right)\left(x-3\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x^{2}+2x-15\right)=0$$

Hei kimi i te tauaro o x^{2}+2x-15, kimihia te tauaro o ia taurangi.

$$2x^{2}+17x+35-x^{2}-2x+15=0$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+17x+35-2x+15=0$$

Pahekotia te 17x me -2x, ka 15x.

$$x^{2}+15x+35+15=0$$

Tāpirihia te 35 ki te 15, ka 50.

$$x^{2}+15x+50=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+15x+50 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=15 ab=50$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 50.

$$1,50 2,25 5,10$$

Tātaihia te tapeke mō ia takirua.

$$1+50=51 2+25=27 5+10=15$$

Ko te otinga te takirua ka hoatu i te tapeke 15.

$$a=5 b=10$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x+5\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x+5=0 me te x+10=0.

$$x=-5 x=-10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te 2x+7 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x+5\right)\left(x-3\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x^{2}+2x-15\right)=0$$

Hei kimi i te tauaro o x^{2}+2x-15, kimihia te tauaro o ia taurangi.

$$2x^{2}+17x+35-x^{2}-2x+15=0$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+17x+35-2x+15=0$$

Pahekotia te 17x me -2x, ka 15x.

$$x^{2}+15x+35+15=0$$

Tāpirihia te 35 ki te 15, ka 50.

$$x^{2}+15x+50=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+50. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=15 ab=1\times 50=50$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 50.

$$1,50 2,25 5,10$$

Tātaihia te tapeke mō ia takirua.

$$1+50=51 2+25=27 5+10=15$$

Ko te otinga te takirua ka hoatu i te tapeke 15.

$$a=5 b=10$$

Tuhia anō te x^{2}+15x+50 hei \left(x^{2}+5x\right)+\left(10x+50\right).

$$\left(x^{2}+5x\right)+\left(10x+50\right)$$

Tauwehea te x i te tuatahi me te 10 i te rōpū tuarua.

$$x\left(x+5\right)+10\left(x+5\right)$$

Whakatauwehea atu te kīanga pātahi x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+5\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x+5=0 me te x+10=0.

$$x=-5 x=-10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te 2x+7 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x+5\right)\left(x-3\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x^{2}+2x-15\right)=0$$

Hei kimi i te tauaro o x^{2}+2x-15, kimihia te tauaro o ia taurangi.

$$2x^{2}+17x+35-x^{2}-2x+15=0$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+17x+35-2x+15=0$$

Pahekotia te 17x me -2x, ka 15x.

$$x^{2}+15x+35+15=0$$

Tāpirihia te 35 ki te 15, ka 50.

$$x^{2}+15x+50=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 15 mō b, me 50 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 50}}{2}$$

Pūrua 15.

$$x=\frac{-15±\sqrt{225-4\times 50}}{2}$$

Whakareatia -4 ki te 50.

$$x=\frac{-15±\sqrt{225-200}}{2}$$

Tāpiri 225 ki te -200.

$$x=\frac{-15±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-15±5}{2}$$

Nā, me whakaoti te whārite x=\frac{-15±5}{2} ina he tāpiri te ±. Tāpiri -15 ki te 5.

$$x=-\frac{10}{2}$$

Whakawehe -10 ki te 2.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-15±5}{2} ina he tango te ±. Tango 5 mai i -15.

$$x=-\frac{20}{2}$$

Whakawehe -20 ki te 2.

$$x=-10$$

Kua oti te whārite te whakatau.

$$x=-5 x=-10$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te 2x+7 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x+5\right)\left(x-3\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}+17x+35-\left(x^{2}+2x-15\right)=0$$

Hei kimi i te tauaro o x^{2}+2x-15, kimihia te tauaro o ia taurangi.

$$2x^{2}+17x+35-x^{2}-2x+15=0$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+17x+35-2x+15=0$$

Pahekotia te 17x me -2x, ka 15x.

$$x^{2}+15x+35+15=0$$

Tāpirihia te 35 ki te 15, ka 50.

$$x^{2}+15x+50=0$$

Tangohia te 50 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}+15x=-50$$

Whakawehea te 15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{2}. Nā, tāpiria te pūrua o te \frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=-50+\left(\frac{15}{2}\right)^{2}$$

Pūruatia \frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+15x+\frac{225}{4}=-50+\frac{225}{4}$$

Tāpiri -50 ki te \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{25}{4}$$

Tauwehea x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$x+\frac{15}{2}=\frac{5}{2} x+\frac{15}{2}=-\frac{5}{2}$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$x=-5 x=-10$$