Whakaoti i te sqrt{x^2+4}+sqrt{left(12-xright)^2+9}=13

$$\sqrt{ { x }^{ 2 } +4 } + \sqrt{ { \left(12-x \right) }^{ 2 } +9 } =13$$

Whakaoti mō x

$$x = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

Ngā Upane Otinga

Me tango \sqrt{\left(12-x\right)^{2}+9} mai i ngā taha e rua o te whārite.

$$\sqrt{x^{2}+4}=13-\sqrt{\left(12-x\right)^{2}+9}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(12-x\right)^{2}.

$$\sqrt{x^{2}+4}=13-\sqrt{144-24x+x^{2}+9}$$

Tāpirihia te 144 ki te 9, ka 153.

$$\sqrt{x^{2}+4}=13-\sqrt{153-24x+x^{2}}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{x^{2}+4}\right)^{2}=\left(13-\sqrt{153-24x+x^{2}}\right)^{2}$$

Tātaihia te \sqrt{x^{2}+4} mā te pū o 2, kia riro ko x^{2}+4.

$$x^{2}+4=\left(13-\sqrt{153-24x+x^{2}}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(13-\sqrt{153-24x+x^{2}}\right)^{2}.

$$x^{2}+4=169-26\sqrt{153-24x+x^{2}}+\left(\sqrt{153-24x+x^{2}}\right)^{2}$$

Tātaihia te \sqrt{153-24x+x^{2}} mā te pū o 2, kia riro ko 153-24x+x^{2}.

$$x^{2}+4=169-26\sqrt{153-24x+x^{2}}+153-24x+x^{2}$$

Tāpirihia te 169 ki te 153, ka 322.

$$x^{2}+4=322-26\sqrt{153-24x+x^{2}}-24x+x^{2}$$

Me tango 322-24x+x^{2} mai i ngā taha e rua o te whārite.

$$x^{2}+4-\left(322-24x+x^{2}\right)=-26\sqrt{153-24x+x^{2}}$$

Hei kimi i te tauaro o 322-24x+x^{2}, kimihia te tauaro o ia taurangi.

$$x^{2}+4-322+24x-x^{2}=-26\sqrt{153-24x+x^{2}}$$

Tangohia te 322 i te 4, ka -318.

$$x^{2}-318+24x-x^{2}=-26\sqrt{153-24x+x^{2}}$$

Pahekotia te x^{2} me -x^{2}, ka 0.

$$-318+24x=-26\sqrt{153-24x+x^{2}}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(-318+24x\right)^{2}=\left(-26\sqrt{153-24x+x^{2}}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(-318+24x\right)^{2}.

$$101124-15264x+576x^{2}=\left(-26\sqrt{153-24x+x^{2}}\right)^{2}$$

Whakarohaina te \left(-26\sqrt{153-24x+x^{2}}\right)^{2}.

$$101124-15264x+576x^{2}=\left(-26\right)^{2}\left(\sqrt{153-24x+x^{2}}\right)^{2}$$

Tātaihia te -26 mā te pū o 2, kia riro ko 676.

$$101124-15264x+576x^{2}=676\left(\sqrt{153-24x+x^{2}}\right)^{2}$$

Tātaihia te \sqrt{153-24x+x^{2}} mā te pū o 2, kia riro ko 153-24x+x^{2}.

$$101124-15264x+576x^{2}=676\left(153-24x+x^{2}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 676 ki te 153-24x+x^{2}.

$$101124-15264x+576x^{2}=103428-16224x+676x^{2}$$

Tangohia te 103428 mai i ngā taha e rua.

$$101124-15264x+576x^{2}-103428=-16224x+676x^{2}$$

Tangohia te 103428 i te 101124, ka -2304.

$$-2304-15264x+576x^{2}=-16224x+676x^{2}$$

Me tāpiri te 16224x ki ngā taha e rua.

$$-2304-15264x+576x^{2}+16224x=676x^{2}$$

Pahekotia te -15264x me 16224x, ka 960x.

$$-2304+960x+576x^{2}=676x^{2}$$

Tangohia te 676x^{2} mai i ngā taha e rua.

$$-2304+960x+576x^{2}-676x^{2}=0$$

Pahekotia te 576x^{2} me -676x^{2}, ka -100x^{2}.

$$-2304+960x-100x^{2}=0$$

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.

$$-100x^{2}+960x-2304=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -100 mō a, 960 mō b, me -2304 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-960±\sqrt{960^{2}-4\left(-100\right)\left(-2304\right)}}{2\left(-100\right)}$$

Pūrua 960.

$$x=\frac{-960±\sqrt{921600-4\left(-100\right)\left(-2304\right)}}{2\left(-100\right)}$$

Whakareatia -4 ki te -100.

$$x=\frac{-960±\sqrt{921600+400\left(-2304\right)}}{2\left(-100\right)}$$

Whakareatia 400 ki te -2304.

$$x=\frac{-960±\sqrt{921600-921600}}{2\left(-100\right)}$$

Tāpiri 921600 ki te -921600.

$$x=\frac{-960±\sqrt{0}}{2\left(-100\right)}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{960}{2\left(-100\right)}$$

Whakareatia 2 ki te -100.

$$x=-\frac{960}{-200}$$

Whakahekea te hautanga \frac{-960}{-200} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 40.

$$x=\frac{24}{5}$$

Whakakapia te \frac{24}{5} mō te x i te whārite \sqrt{x^{2}+4}+\sqrt{\left(12-x\right)^{2}+9}=13.

$$\sqrt{\left(\frac{24}{5}\right)^{2}+4}+\sqrt{\left(12-\frac{24}{5}\right)^{2}+9}=13$$

Whakarūnātia. Ko te uara x=\frac{24}{5} kua ngata te whārite.

$$13=13$$

Ko te whārite \sqrt{x^{2}+4}=-\sqrt{x^{2}-24x+153}+13 he rongoā ahurei.

$$x=\frac{24}{5}$$