Whakaoti i te (x-159)(359-x)=20(x+87)

$$(x-159)(359-x)=20(x+87)$$

Whakaoti mō x

$$x=2\sqrt{795}+249\approx 305.391488719$$

$$x=249-2\sqrt{795}\approx 192.608511281$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-159 ki te 359-x ka whakakotahi i ngā kupu rite.

$$518x-x^{2}-57081=20\left(x+87\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 20 ki te x+87.

$$518x-x^{2}-57081=20x+1740$$

Tangohia te 20x mai i ngā taha e rua.

$$518x-x^{2}-57081-20x=1740$$

Pahekotia te 518x me -20x, ka 498x.

$$498x-x^{2}-57081=1740$$

Tangohia te 1740 mai i ngā taha e rua.

$$498x-x^{2}-57081-1740=0$$

Tangohia te 1740 i te -57081, ka -58821.

$$498x-x^{2}-58821=0$$

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.

$$-x^{2}+498x-58821=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 498 mō b, me -58821 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-498±\sqrt{498^{2}-4\left(-1\right)\left(-58821\right)}}{2\left(-1\right)}$$

Pūrua 498.

$$x=\frac{-498±\sqrt{248004-4\left(-1\right)\left(-58821\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-498±\sqrt{248004+4\left(-58821\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -58821.

$$x=\frac{-498±\sqrt{248004-235284}}{2\left(-1\right)}$$

Tāpiri 248004 ki te -235284.

$$x=\frac{-498±\sqrt{12720}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 12720.

$$x=\frac{-498±4\sqrt{795}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-498±4\sqrt{795}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-498±4\sqrt{795}}{-2} ina he tāpiri te ±. Tāpiri -498 ki te 4\sqrt{795}.

$$x=\frac{4\sqrt{795}-498}{-2}$$

Whakawehe -498+4\sqrt{795} ki te -2.

$$x=249-2\sqrt{795}$$

Nā, me whakaoti te whārite x=\frac{-498±4\sqrt{795}}{-2} ina he tango te ±. Tango 4\sqrt{795} mai i -498.

$$x=\frac{-4\sqrt{795}-498}{-2}$$

Whakawehe -498-4\sqrt{795} ki te -2.

$$x=2\sqrt{795}+249$$

Kua oti te whārite te whakatau.

$$x=249-2\sqrt{795} x=2\sqrt{795}+249$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-159 ki te 359-x ka whakakotahi i ngā kupu rite.

$$518x-x^{2}-57081=20\left(x+87\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 20 ki te x+87.

$$518x-x^{2}-57081=20x+1740$$

Tangohia te 20x mai i ngā taha e rua.

$$518x-x^{2}-57081-20x=1740$$

Pahekotia te 518x me -20x, ka 498x.

$$498x-x^{2}-57081=1740$$

Me tāpiri te 57081 ki ngā taha e rua.

$$498x-x^{2}=1740+57081$$

Tāpirihia te 1740 ki te 57081, ka 58821.

$$498x-x^{2}=58821$$

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.

$$-x^{2}+498x=58821$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+498x}{-1}=\frac{58821}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{498}{-1}x=\frac{58821}{-1}$$

Whakawehe 498 ki te -1.

$$x^{2}-498x=\frac{58821}{-1}$$

Whakawehe 58821 ki te -1.

$$x^{2}-498x=-58821$$

Whakawehea te -498, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -249. Nā, tāpiria te pūrua o te -249 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-498x+\left(-249\right)^{2}=-58821+\left(-249\right)^{2}$$

Pūrua -249.

$$x^{2}-498x+62001=-58821+62001$$

Tāpiri -58821 ki te 62001.

$$x^{2}-498x+62001=3180$$

Tauwehea te x^{2}-498x+62001. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-249\right)^{2}=3180$$

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

$$\sqrt{\left(x-249\right)^{2}}=\sqrt{3180}$$

Whakarūnātia.

$$x-249=2\sqrt{795} x-249=-2\sqrt{795}$$

Me tāpiri 249 ki ngā taha e rua o te whārite.

$$x=2\sqrt{795}+249 x=249-2\sqrt{795}$$