Whakaoti i te 30-(x+1)-(16-x)=sqrt{(x+1)^2+(16-x)^2}

$$30 - ( x + 1 ) - ( 16 - x ) = \sqrt { ( x + 1 ) ^ { 2 } + ( 16 - x ) ^ { 2 } }$$

Whakaoti mō x

$$x=11$$

$$x=4$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

$$\left(30-\left(x+1\right)-\left(16-x\right)\right)^{2}=\left(\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}\right)^{2}$$

Hei kimi i te tauaro o x+1, kimihia te tauaro o ia taurangi.

$$\left(30-x-1-\left(16-x\right)\right)^{2}=\left(\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}\right)^{2}$$

Tangohia te 1 i te 30, ka 29.

$$\left(29-x-\left(16-x\right)\right)^{2}=\left(\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}\right)^{2}$$

Hei kimi i te tauaro o 16-x, kimihia te tauaro o ia taurangi.

$$\left(29-x-16+x\right)^{2}=\left(\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}\right)^{2}$$

Tangohia te 16 i te 29, ka 13.

$$\left(13-x+x\right)^{2}=\left(\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}\right)^{2}$$

Pahekotia te -x me x, ka 0.

$$13^{2}=\left(\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}\right)^{2}$$

Tātaihia te 13 mā te pū o 2, kia riro ko 169.

$$169=\left(\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$169=\left(\sqrt{x^{2}+2x+1+\left(16-x\right)^{2}}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(16-x\right)^{2}.

$$169=\left(\sqrt{x^{2}+2x+1+256-32x+x^{2}}\right)^{2}$$

Tāpirihia te 1 ki te 256, ka 257.

$$169=\left(\sqrt{x^{2}+2x+257-32x+x^{2}}\right)^{2}$$

Pahekotia te 2x me -32x, ka -30x.

$$169=\left(\sqrt{x^{2}-30x+257+x^{2}}\right)^{2}$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$169=\left(\sqrt{2x^{2}-30x+257}\right)^{2}$$

Tātaihia te \sqrt{2x^{2}-30x+257} mā te pū o 2, kia riro ko 2x^{2}-30x+257.

$$169=2x^{2}-30x+257$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2x^{2}-30x+257=169$$

Tangohia te 169 mai i ngā taha e rua.

$$2x^{2}-30x+257-169=0$$

Tangohia te 169 i te 257, ka 88.

$$2x^{2}-30x+88=0$$

Whakawehea ngā taha e rua ki te 2.

$$x^{2}-15x+44=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+44. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-15 ab=1\times 44=44$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 44.

$$-1,-44 -2,-22 -4,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

Ko te otinga te takirua ka hoatu i te tapeke -15.

$$a=-11 b=-4$$

Tuhia anō te x^{2}-15x+44 hei \left(x^{2}-11x\right)+\left(-4x+44\right).

$$\left(x^{2}-11x\right)+\left(-4x+44\right)$$

Tauwehea te x i te tuatahi me te -4 i te rōpū tuarua.

$$x\left(x-11\right)-4\left(x-11\right)$$

Whakatauwehea atu te kīanga pātahi x-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-11\right)\left(x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-11=0 me te x-4=0.

$$x=11 x=4$$

Whakakapia te 11 mō te x i te whārite 30-\left(x+1\right)-\left(16-x\right)=\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}.

$$30-\left(11+1\right)-\left(16-11\right)=\sqrt{\left(11+1\right)^{2}+\left(16-11\right)^{2}}$$

Whakarūnātia. Ko te uara x=11 kua ngata te whārite.

$$13=13$$

Whakakapia te 4 mō te x i te whārite 30-\left(x+1\right)-\left(16-x\right)=\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}.

$$30-\left(4+1\right)-\left(16-4\right)=\sqrt{\left(4+1\right)^{2}+\left(16-4\right)^{2}}$$

Whakarūnātia. Ko te uara x=4 kua ngata te whārite.

$$13=13$$

Rārangihia ngā rongoā katoa o -\left(x+1\right)-\left(16-x\right)+30=\sqrt{\left(x+1\right)^{2}+\left(16-x\right)^{2}}.

$$x=11 x=4$$