Whakaoti i te (15-x)(5x+500)=4250

$$( 15 - x ) ( 5 x + 500 ) = 4250$$

Whakaoti mō x

$$x = \frac{5 \sqrt{393} - 85}{2} \approx 7.060569004$$

$$x=\frac{-5\sqrt{393}-85}{2}\approx -92.060569004$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 15-x ki te 5x+500 ka whakakotahi i ngā kupu rite.

$$-425x+7500-5x^{2}=4250$$

Tangohia te 4250 mai i ngā taha e rua.

$$-425x+7500-5x^{2}-4250=0$$

Tangohia te 4250 i te 7500, ka 3250.

$$-425x+3250-5x^{2}=0$$

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.

$$-5x^{2}-425x+3250=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, -425 mō b, me 3250 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-425\right)±\sqrt{\left(-425\right)^{2}-4\left(-5\right)\times 3250}}{2\left(-5\right)}$$

Pūrua -425.

$$x=\frac{-\left(-425\right)±\sqrt{180625-4\left(-5\right)\times 3250}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$x=\frac{-\left(-425\right)±\sqrt{180625+20\times 3250}}{2\left(-5\right)}$$

Whakareatia 20 ki te 3250.

$$x=\frac{-\left(-425\right)±\sqrt{180625+65000}}{2\left(-5\right)}$$

Tāpiri 180625 ki te 65000.

$$x=\frac{-\left(-425\right)±\sqrt{245625}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te 245625.

$$x=\frac{-\left(-425\right)±25\sqrt{393}}{2\left(-5\right)}$$

Ko te tauaro o -425 ko 425.

$$x=\frac{425±25\sqrt{393}}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$x=\frac{425±25\sqrt{393}}{-10}$$

Nā, me whakaoti te whārite x=\frac{425±25\sqrt{393}}{-10} ina he tāpiri te ±. Tāpiri 425 ki te 25\sqrt{393}.

$$x=\frac{25\sqrt{393}+425}{-10}$$

Whakawehe 425+25\sqrt{393} ki te -10.

$$x=\frac{-5\sqrt{393}-85}{2}$$

Nā, me whakaoti te whārite x=\frac{425±25\sqrt{393}}{-10} ina he tango te ±. Tango 25\sqrt{393} mai i 425.

$$x=\frac{425-25\sqrt{393}}{-10}$$

Whakawehe 425-25\sqrt{393} ki te -10.

$$x=\frac{5\sqrt{393}-85}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-5\sqrt{393}-85}{2} x=\frac{5\sqrt{393}-85}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 15-x ki te 5x+500 ka whakakotahi i ngā kupu rite.

$$-425x+7500-5x^{2}=4250$$

Tangohia te 7500 mai i ngā taha e rua.

$$-425x-5x^{2}=4250-7500$$

Tangohia te 7500 i te 4250, ka -3250.

$$-425x-5x^{2}=-3250$$

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.

$$-5x^{2}-425x=-3250$$

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5x^{2}-425x}{-5}=-\frac{3250}{-5}$$

Mā te whakawehe ki te -5 ka wetekia te whakareanga ki te -5.

$$x^{2}+\left(-\frac{425}{-5}\right)x=-\frac{3250}{-5}$$

Whakawehe -425 ki te -5.

$$x^{2}+85x=-\frac{3250}{-5}$$

Whakawehe -3250 ki te -5.

$$x^{2}+85x=650$$

Whakawehea te 85, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{85}{2}. Nā, tāpiria te pūrua o te \frac{85}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+85x+\left(\frac{85}{2}\right)^{2}=650+\left(\frac{85}{2}\right)^{2}$$

Pūruatia \frac{85}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+85x+\frac{7225}{4}=650+\frac{7225}{4}$$

Tāpiri 650 ki te \frac{7225}{4}.

$$x^{2}+85x+\frac{7225}{4}=\frac{9825}{4}$$

Tauwehea te x^{2}+85x+\frac{7225}{4}. 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}.

$$\left(x+\frac{85}{2}\right)^{2}=\frac{9825}{4}$$

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

$$\sqrt{\left(x+\frac{85}{2}\right)^{2}}=\sqrt{\frac{9825}{4}}$$

Whakarūnātia.

$$x+\frac{85}{2}=\frac{5\sqrt{393}}{2} x+\frac{85}{2}=-\frac{5\sqrt{393}}{2}$$

Me tango \frac{85}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{393}-85}{2} x=\frac{-5\sqrt{393}-85}{2}$$