Whakaoti i te left(2x-40right)left(3x-50right)left(30+100right)+2000*100=642000

$$\left( 2x-40 \right) \left( 3x-50 \right) \left( 30+100 \right) +2000 \cdot 100 = 642000$$

Whakaoti mō x

$$x = \frac{5 \sqrt{205} + 55}{3} \approx 42.196368439$$

$$x=\frac{55-5\sqrt{205}}{3}\approx -5.529701772$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tāpirihia te 30 ki te 100, ka 130.

$$\left(2x-40\right)\left(3x-50\right)\times 130+2000\times 100=642000$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-40 ki te 3x-50 ka whakakotahi i ngā kupu rite.

$$\left(6x^{2}-220x+2000\right)\times 130+2000\times 100=642000$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x^{2}-220x+2000 ki te 130.

$$780x^{2}-28600x+260000+2000\times 100=642000$$

Whakareatia te 2000 ki te 100, ka 200000.

$$780x^{2}-28600x+260000+200000=642000$$

Tāpirihia te 260000 ki te 200000, ka 460000.

$$780x^{2}-28600x+460000=642000$$

Tangohia te 642000 mai i ngā taha e rua.

$$780x^{2}-28600x+460000-642000=0$$

Tangohia te 642000 i te 460000, ka -182000.

$$780x^{2}-28600x-182000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 780 mō a, -28600 mō b, me -182000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28600\right)±\sqrt{\left(-28600\right)^{2}-4\times 780\left(-182000\right)}}{2\times 780}$$

Pūrua -28600.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-4\times 780\left(-182000\right)}}{2\times 780}$$

Whakareatia -4 ki te 780.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-3120\left(-182000\right)}}{2\times 780}$$

Whakareatia -3120 ki te -182000.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000+567840000}}{2\times 780}$$

Tāpiri 817960000 ki te 567840000.

$$x=\frac{-\left(-28600\right)±\sqrt{1385800000}}{2\times 780}$$

Tuhia te pūtakerua o te 1385800000.

$$x=\frac{-\left(-28600\right)±2600\sqrt{205}}{2\times 780}$$

Ko te tauaro o -28600 ko 28600.

$$x=\frac{28600±2600\sqrt{205}}{2\times 780}$$

Whakareatia 2 ki te 780.

$$x=\frac{28600±2600\sqrt{205}}{1560}$$

Nā, me whakaoti te whārite x=\frac{28600±2600\sqrt{205}}{1560} ina he tāpiri te ±. Tāpiri 28600 ki te 2600\sqrt{205}.

$$x=\frac{2600\sqrt{205}+28600}{1560}$$

Whakawehe 28600+2600\sqrt{205} ki te 1560.

$$x=\frac{5\sqrt{205}+55}{3}$$

Nā, me whakaoti te whārite x=\frac{28600±2600\sqrt{205}}{1560} ina he tango te ±. Tango 2600\sqrt{205} mai i 28600.

$$x=\frac{28600-2600\sqrt{205}}{1560}$$

Whakawehe 28600-2600\sqrt{205} ki te 1560.

$$x=\frac{55-5\sqrt{205}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{5\sqrt{205}+55}{3} x=\frac{55-5\sqrt{205}}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tāpirihia te 30 ki te 100, ka 130.

$$\left(2x-40\right)\left(3x-50\right)\times 130+2000\times 100=642000$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-40 ki te 3x-50 ka whakakotahi i ngā kupu rite.

$$\left(6x^{2}-220x+2000\right)\times 130+2000\times 100=642000$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x^{2}-220x+2000 ki te 130.

$$780x^{2}-28600x+260000+2000\times 100=642000$$

Whakareatia te 2000 ki te 100, ka 200000.

$$780x^{2}-28600x+260000+200000=642000$$

Tāpirihia te 260000 ki te 200000, ka 460000.

$$780x^{2}-28600x+460000=642000$$

Tangohia te 460000 mai i ngā taha e rua.

$$780x^{2}-28600x=642000-460000$$

Tangohia te 460000 i te 642000, ka 182000.

$$780x^{2}-28600x=182000$$

Whakawehea ngā taha e rua ki te 780.

$$\frac{780x^{2}-28600x}{780}=\frac{182000}{780}$$

Mā te whakawehe ki te 780 ka wetekia te whakareanga ki te 780.

$$x^{2}+\left(-\frac{28600}{780}\right)x=\frac{182000}{780}$$

Whakahekea te hautanga \frac{-28600}{780} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 260.

$$x^{2}-\frac{110}{3}x=\frac{182000}{780}$$

Whakahekea te hautanga \frac{182000}{780} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 260.

$$x^{2}-\frac{110}{3}x=\frac{700}{3}$$

Whakawehea te -\frac{110}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{55}{3}. Nā, tāpiria te pūrua o te -\frac{55}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{110}{3}x+\left(-\frac{55}{3}\right)^{2}=\frac{700}{3}+\left(-\frac{55}{3}\right)^{2}$$

Pūruatia -\frac{55}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{700}{3}+\frac{3025}{9}$$

Tāpiri \frac{700}{3} ki te \frac{3025}{9} 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.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{5125}{9}$$

Tauwehea x^{2}-\frac{110}{3}x+\frac{3025}{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}.

$$\left(x-\frac{55}{3}\right)^{2}=\frac{5125}{9}$$

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

$$\sqrt{\left(x-\frac{55}{3}\right)^{2}}=\sqrt{\frac{5125}{9}}$$

Whakarūnātia.

$$x-\frac{55}{3}=\frac{5\sqrt{205}}{3} x-\frac{55}{3}=-\frac{5\sqrt{205}}{3}$$

Me tāpiri \frac{55}{3} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{205}+55}{3} x=\frac{55-5\sqrt{205}}{3}$$