Whakaoti i te 25left(4+xright)^2+7(5-x)(5+x)=295-45x^2

$$25 { \left(4+x \right) }^{ 2 } +7(5-x)(5+x)=295-45 { x }^{ 2 }$$

Whakaoti mō x (complex solution)

$$x=\frac{-100+2\sqrt{1910}i}{63}\approx -1.587301587+1.387414183i$$

$$x=\frac{-2\sqrt{1910}i-100}{63}\approx -1.587301587-1.387414183i$$

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

$$25\left(16+8x+x^{2}\right)+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 25 ki te 16+8x+x^{2}.

$$400+200x+25x^{2}+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te 5-x.

$$400+200x+25x^{2}+\left(35-7x\right)\left(5+x\right)=295-45x^{2}$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 35-7x ki te 5+x ka whakakotahi i ngā kupu rite.

$$400+200x+25x^{2}+175-7x^{2}=295-45x^{2}$$

Tāpirihia te 400 ki te 175, ka 575.

$$575+200x+25x^{2}-7x^{2}=295-45x^{2}$$

Pahekotia te 25x^{2} me -7x^{2}, ka 18x^{2}.

$$575+200x+18x^{2}=295-45x^{2}$$

Tangohia te 295 mai i ngā taha e rua.

$$575+200x+18x^{2}-295=-45x^{2}$$

Tangohia te 295 i te 575, ka 280.

$$280+200x+18x^{2}=-45x^{2}$$

Me tāpiri te 45x^{2} ki ngā taha e rua.

$$280+200x+18x^{2}+45x^{2}=0$$

Pahekotia te 18x^{2} me 45x^{2}, ka 63x^{2}.

$$280+200x+63x^{2}=0$$

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.

$$63x^{2}+200x+280=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 63 mō a, 200 mō b, me 280 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\times 63\times 280}}{2\times 63}$$

Pūrua 200.

$$x=\frac{-200±\sqrt{40000-4\times 63\times 280}}{2\times 63}$$

Whakareatia -4 ki te 63.

$$x=\frac{-200±\sqrt{40000-252\times 280}}{2\times 63}$$

Whakareatia -252 ki te 280.

$$x=\frac{-200±\sqrt{40000-70560}}{2\times 63}$$

Tāpiri 40000 ki te -70560.

$$x=\frac{-200±\sqrt{-30560}}{2\times 63}$$

Tuhia te pūtakerua o te -30560.

$$x=\frac{-200±4\sqrt{1910}i}{2\times 63}$$

Whakareatia 2 ki te 63.

$$x=\frac{-200±4\sqrt{1910}i}{126}$$

Nā, me whakaoti te whārite x=\frac{-200±4\sqrt{1910}i}{126} ina he tāpiri te ±. Tāpiri -200 ki te 4i\sqrt{1910}.

$$x=\frac{-200+4\sqrt{1910}i}{126}$$

Whakawehe -200+4i\sqrt{1910} ki te 126.

$$x=\frac{-100+2\sqrt{1910}i}{63}$$

Nā, me whakaoti te whārite x=\frac{-200±4\sqrt{1910}i}{126} ina he tango te ±. Tango 4i\sqrt{1910} mai i -200.

$$x=\frac{-4\sqrt{1910}i-200}{126}$$

Whakawehe -200-4i\sqrt{1910} ki te 126.

$$x=\frac{-2\sqrt{1910}i-100}{63}$$

Kua oti te whārite te whakatau.

$$x=\frac{-100+2\sqrt{1910}i}{63} x=\frac{-2\sqrt{1910}i-100}{63}$$

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

$$25\left(16+8x+x^{2}\right)+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 25 ki te 16+8x+x^{2}.

$$400+200x+25x^{2}+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te 5-x.

$$400+200x+25x^{2}+\left(35-7x\right)\left(5+x\right)=295-45x^{2}$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 35-7x ki te 5+x ka whakakotahi i ngā kupu rite.

$$400+200x+25x^{2}+175-7x^{2}=295-45x^{2}$$

Tāpirihia te 400 ki te 175, ka 575.

$$575+200x+25x^{2}-7x^{2}=295-45x^{2}$$

Pahekotia te 25x^{2} me -7x^{2}, ka 18x^{2}.

$$575+200x+18x^{2}=295-45x^{2}$$

Me tāpiri te 45x^{2} ki ngā taha e rua.

$$575+200x+18x^{2}+45x^{2}=295$$

Pahekotia te 18x^{2} me 45x^{2}, ka 63x^{2}.

$$575+200x+63x^{2}=295$$

Tangohia te 575 mai i ngā taha e rua.

$$200x+63x^{2}=295-575$$

Tangohia te 575 i te 295, ka -280.

$$200x+63x^{2}=-280$$

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.

$$63x^{2}+200x=-280$$

Whakawehea ngā taha e rua ki te 63.

$$\frac{63x^{2}+200x}{63}=-\frac{280}{63}$$

Mā te whakawehe ki te 63 ka wetekia te whakareanga ki te 63.

$$x^{2}+\frac{200}{63}x=-\frac{280}{63}$$

Whakahekea te hautanga \frac{-280}{63} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 7.

$$x^{2}+\frac{200}{63}x=-\frac{40}{9}$$

Whakawehea te \frac{200}{63}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{100}{63}. Nā, tāpiria te pūrua o te \frac{100}{63} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{200}{63}x+\left(\frac{100}{63}\right)^{2}=-\frac{40}{9}+\left(\frac{100}{63}\right)^{2}$$

Pūruatia \frac{100}{63} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{200}{63}x+\frac{10000}{3969}=-\frac{40}{9}+\frac{10000}{3969}$$

Tāpiri -\frac{40}{9} ki te \frac{10000}{3969} 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.

$$x^{2}+\frac{200}{63}x+\frac{10000}{3969}=-\frac{7640}{3969}$$

Tauwehea x^{2}+\frac{200}{63}x+\frac{10000}{3969}. 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}.

$$\left(x+\frac{100}{63}\right)^{2}=-\frac{7640}{3969}$$

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

$$\sqrt{\left(x+\frac{100}{63}\right)^{2}}=\sqrt{-\frac{7640}{3969}}$$

Whakarūnātia.

$$x+\frac{100}{63}=\frac{2\sqrt{1910}i}{63} x+\frac{100}{63}=-\frac{2\sqrt{1910}i}{63}$$

Me tango \frac{100}{63} mai i ngā taha e rua o te whārite.

$$x=\frac{-100+2\sqrt{1910}i}{63} x=\frac{-2\sqrt{1910}i-100}{63}$$