Whakaoti i te quadtext{36If}frac{sqrt{7}-2}{sqrt{7}+2}=asqrt{7}+b

$$\quad \text { 36 If } \frac { \sqrt { 7 } - 2 } { \sqrt { 7 } + 2 } = a \sqrt { 7 } + b$$

Whakaoti mō I

$$\left\{\begin{matrix}I=\frac{4\sqrt{7}b+11\sqrt{7}a+11b+28a}{108f}\text{, }&f\neq 0\\I\in \mathrm{R}\text{, }&a=-\frac{\sqrt{7}b}{7}\text{ and }f=0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakangāwaritia te tauraro o \frac{\sqrt{7}-2}{\sqrt{7}+2} mā te whakarea i te taurunga me te tauraro ki te \sqrt{7}-2.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{\left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right)}=a\sqrt{7}+b$$

Whakaarohia te \left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{\left(\sqrt{7}\right)^{2}-2^{2}}=a\sqrt{7}+b$$

Pūrua \sqrt{7}. Pūrua 2.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{7-4}=a\sqrt{7}+b$$

Tangohia te 4 i te 7, ka 3.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{3}=a\sqrt{7}+b$$

Whakareatia te \sqrt{7}-2 ki te \sqrt{7}-2, ka \left(\sqrt{7}-2\right)^{2}.

$$36If\times \frac{\left(\sqrt{7}-2\right)^{2}}{3}=a\sqrt{7}+b$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{7}-2\right)^{2}.

$$36If\times \frac{\left(\sqrt{7}\right)^{2}-4\sqrt{7}+4}{3}=a\sqrt{7}+b$$

Ko te pūrua o \sqrt{7} ko 7.

$$36If\times \frac{7-4\sqrt{7}+4}{3}=a\sqrt{7}+b$$

Tāpirihia te 7 ki te 4, ka 11.

$$36If\times \frac{11-4\sqrt{7}}{3}=a\sqrt{7}+b$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te 36 me te 3.

$$12\left(11-4\sqrt{7}\right)If=a\sqrt{7}+b$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 11-4\sqrt{7}.

$$\left(132-48\sqrt{7}\right)If=a\sqrt{7}+b$$

Whakamahia te āhuatanga tohatoha hei whakarea te 132-48\sqrt{7} ki te I.

$$\left(132I-48\sqrt{7}I\right)f=a\sqrt{7}+b$$

Whakamahia te āhuatanga tohatoha hei whakarea te 132I-48\sqrt{7}I ki te f.

$$132If-48\sqrt{7}If=a\sqrt{7}+b$$

Pahekotia ngā kīanga tau katoa e whai ana i te I.

$$\left(132f-48\sqrt{7}f\right)I=a\sqrt{7}+b$$

He hanga arowhānui tō te whārite.

$$\left(-48\sqrt{7}f+132f\right)I=\sqrt{7}a+b$$

Whakawehea ngā taha e rua ki te 132f-48\sqrt{7}f.

$$\frac{\left(-48\sqrt{7}f+132f\right)I}{-48\sqrt{7}f+132f}=\frac{\sqrt{7}a+b}{-48\sqrt{7}f+132f}$$

Mā te whakawehe ki te 132f-48\sqrt{7}f ka wetekia te whakareanga ki te 132f-48\sqrt{7}f.

$$I=\frac{\sqrt{7}a+b}{-48\sqrt{7}f+132f}$$

Whakawehe a\sqrt{7}+b ki te 132f-48\sqrt{7}f.

$$I=\frac{\left(4\sqrt{7}+11\right)\left(\sqrt{7}a+b\right)}{108f}$$

Whakaoti mō a

$$a=-\frac{\sqrt{7}\left(48\sqrt{7}If-132If+b\right)}{7}$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakangāwaritia te tauraro o \frac{\sqrt{7}-2}{\sqrt{7}+2} mā te whakarea i te taurunga me te tauraro ki te \sqrt{7}-2.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{\left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right)}=a\sqrt{7}+b$$

Whakaarohia te \left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{\left(\sqrt{7}\right)^{2}-2^{2}}=a\sqrt{7}+b$$

Pūrua \sqrt{7}. Pūrua 2.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{7-4}=a\sqrt{7}+b$$

Tangohia te 4 i te 7, ka 3.

$$36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{3}=a\sqrt{7}+b$$

Whakareatia te \sqrt{7}-2 ki te \sqrt{7}-2, ka \left(\sqrt{7}-2\right)^{2}.

$$36If\times \frac{\left(\sqrt{7}-2\right)^{2}}{3}=a\sqrt{7}+b$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{7}-2\right)^{2}.

$$36If\times \frac{\left(\sqrt{7}\right)^{2}-4\sqrt{7}+4}{3}=a\sqrt{7}+b$$

Ko te pūrua o \sqrt{7} ko 7.

$$36If\times \frac{7-4\sqrt{7}+4}{3}=a\sqrt{7}+b$$

Tāpirihia te 7 ki te 4, ka 11.

$$36If\times \frac{11-4\sqrt{7}}{3}=a\sqrt{7}+b$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te 36 me te 3.

$$12\left(11-4\sqrt{7}\right)If=a\sqrt{7}+b$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 11-4\sqrt{7}.

$$\left(132-48\sqrt{7}\right)If=a\sqrt{7}+b$$

Whakamahia te āhuatanga tohatoha hei whakarea te 132-48\sqrt{7} ki te I.

$$\left(132I-48\sqrt{7}I\right)f=a\sqrt{7}+b$$

Whakamahia te āhuatanga tohatoha hei whakarea te 132I-48\sqrt{7}I ki te f.

$$132If-48\sqrt{7}If=a\sqrt{7}+b$$

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

$$a\sqrt{7}+b=132If-48\sqrt{7}If$$

Tangohia te b mai i ngā taha e rua.

$$a\sqrt{7}=132If-48\sqrt{7}If-b$$

He hanga arowhānui tō te whārite.

$$\sqrt{7}a=-48\sqrt{7}If+132If-b$$

Whakawehea ngā taha e rua ki te \sqrt{7}.

$$\frac{\sqrt{7}a}{\sqrt{7}}=\frac{-48\sqrt{7}If+132If-b}{\sqrt{7}}$$

Mā te whakawehe ki te \sqrt{7} ka wetekia te whakareanga ki te \sqrt{7}.

$$a=\frac{-48\sqrt{7}If+132If-b}{\sqrt{7}}$$

Whakawehe -b+132fI-48\sqrt{7}fI ki te \sqrt{7}.

$$a=\frac{\sqrt{7}\left(-48\sqrt{7}If+132If-b\right)}{7}$$