Whakaoti i te (40-m)(20+2m)=120

$$( 40 - m ) ( 20 + 2 m ) = 120$$

Whakaoti mō m

$$m=\sqrt{565}+15\approx 38.769728648$$

$$m=15-\sqrt{565}\approx -8.769728648$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-m ki te 20+2m ka whakakotahi i ngā kupu rite.

$$800+60m-2m^{2}=120$$

Tangohia te 120 mai i ngā taha e rua.

$$800+60m-2m^{2}-120=0$$

Tangohia te 120 i te 800, ka 680.

$$680+60m-2m^{2}=0$$

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.

$$-2m^{2}+60m+680=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 60 mō b, me 680 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-60±\sqrt{60^{2}-4\left(-2\right)\times 680}}{2\left(-2\right)}$$

Pūrua 60.

$$m=\frac{-60±\sqrt{3600-4\left(-2\right)\times 680}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$m=\frac{-60±\sqrt{3600+8\times 680}}{2\left(-2\right)}$$

Whakareatia 8 ki te 680.

$$m=\frac{-60±\sqrt{3600+5440}}{2\left(-2\right)}$$

Tāpiri 3600 ki te 5440.

$$m=\frac{-60±\sqrt{9040}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 9040.

$$m=\frac{-60±4\sqrt{565}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$m=\frac{-60±4\sqrt{565}}{-4}$$

Nā, me whakaoti te whārite m=\frac{-60±4\sqrt{565}}{-4} ina he tāpiri te ±. Tāpiri -60 ki te 4\sqrt{565}.

$$m=\frac{4\sqrt{565}-60}{-4}$$

Whakawehe -60+4\sqrt{565} ki te -4.

$$m=15-\sqrt{565}$$

Nā, me whakaoti te whārite m=\frac{-60±4\sqrt{565}}{-4} ina he tango te ±. Tango 4\sqrt{565} mai i -60.

$$m=\frac{-4\sqrt{565}-60}{-4}$$

Whakawehe -60-4\sqrt{565} ki te -4.

$$m=\sqrt{565}+15$$

Kua oti te whārite te whakatau.

$$m=15-\sqrt{565} m=\sqrt{565}+15$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 40-m ki te 20+2m ka whakakotahi i ngā kupu rite.

$$800+60m-2m^{2}=120$$

Tangohia te 800 mai i ngā taha e rua.

$$60m-2m^{2}=120-800$$

Tangohia te 800 i te 120, ka -680.

$$60m-2m^{2}=-680$$

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.

$$-2m^{2}+60m=-680$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2m^{2}+60m}{-2}=-\frac{680}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$m^{2}+\frac{60}{-2}m=-\frac{680}{-2}$$

Whakawehe 60 ki te -2.

$$m^{2}-30m=-\frac{680}{-2}$$

Whakawehe -680 ki te -2.

$$m^{2}-30m=340$$

Whakawehea te -30, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -15. Nā, tāpiria te pūrua o te -15 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$m^{2}-30m+\left(-15\right)^{2}=340+\left(-15\right)^{2}$$

Pūrua -15.

$$m^{2}-30m+225=340+225$$

Tāpiri 340 ki te 225.

$$m^{2}-30m+225=565$$

Tauwehea m^{2}-30m+225. 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}.

$$\left(m-15\right)^{2}=565$$

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

$$\sqrt{\left(m-15\right)^{2}}=\sqrt{565}$$

Whakarūnātia.

$$m-15=\sqrt{565} m-15=-\sqrt{565}$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$m=\sqrt{565}+15 m=15-\sqrt{565}$$