Whakaoti i te 15z^2+34z+15>0

$$15 z ^ { 2 } + 34 z + 15 > 0$$

Whakaoti mō z

$$z\in \left(-\infty,-\frac{5}{3}\right)\cup \left(-\frac{3}{5},\infty\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Kia whakaotia te koreōrite, me tauwehe te taha mauī. Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$15z^{2}+34z+15=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 15 mō te a, te 34 mō te b, me te 15 mō te c i te ture pūrua.

$$z=\frac{-34±\sqrt{34^{2}-4\times 15\times 15}}{2\times 15}$$

Mahia ngā tātaitai.

$$z=\frac{-34±16}{30}$$

Whakaotia te whārite z=\frac{-34±16}{30} ina he tōrunga te ±, ina he tōraro te ±.

$$z=-\frac{3}{5} z=-\frac{5}{3}$$

Tuhia anō te koreōrite mā te whakamahi i ngā otinga i whiwhi.

$$15\left(z+\frac{3}{5}\right)\left(z+\frac{5}{3}\right)>0$$

Kia tōrunga te otinga, me tōraro tahi te z+\frac{3}{5} me te z+\frac{5}{3}, me tōrunga tahi rānei. Whakaarohia te tauira ina he tōraro tahi te z+\frac{3}{5} me te z+\frac{5}{3}.

$$z+\frac{3}{5}<0 z+\frac{5}{3}<0$$

Te otinga e whakaea i ngā koreōrite e rua ko z<-\frac{5}{3}.

$$z<-\frac{5}{3}$$

Whakaarohia te tauira ina he tōrunga tahi te z+\frac{3}{5} me te z+\frac{5}{3}.

$$z+\frac{5}{3}>0 z+\frac{3}{5}>0$$

Te otinga e whakaea i ngā koreōrite e rua ko z>-\frac{3}{5}.

$$z>-\frac{3}{5}$$

Ko te otinga whakamutunga ko te whakakotahi i ngā otinga kua whiwhi.

$$z<-\frac{5}{3}\text{; }z>-\frac{3}{5}$$