\quad \text { 36. If } \frac { \sqrt { 7 } - 2 } { \sqrt { 7 } + 2 } = a \sqrt { 7 } + b
Solve for I (complex solution)
\left\{\begin{matrix}I=\frac{4\sqrt{7}b+11\sqrt{7}a+11b+28a}{108f}\text{, }&f\neq 0\\I\in \mathrm{C}\text{, }&a=-\frac{\sqrt{7}b}{7}\text{ and }f=0\end{matrix}\right.
Solve for 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.
Solve for a
a=-\frac{\sqrt{7}\left(48\sqrt{7}If-132If+b\right)}{7}
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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
Rationalize the denominator of \frac{\sqrt{7}-2}{\sqrt{7}+2} by multiplying numerator and denominator by \sqrt{7}-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
Consider \left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right). Multiplication can be transformed into difference of squares using the rule: \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)}{7-4}=a\sqrt{7}+b
Square \sqrt{7}. Square 2.
36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{3}=a\sqrt{7}+b
Subtract 4 from 7 to get 3.
36If\times \frac{\left(\sqrt{7}-2\right)^{2}}{3}=a\sqrt{7}+b
Multiply \sqrt{7}-2 and \sqrt{7}-2 to get \left(\sqrt{7}-2\right)^{2}.
36If\times \frac{\left(\sqrt{7}\right)^{2}-4\sqrt{7}+4}{3}=a\sqrt{7}+b
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{7}-2\right)^{2}.
36If\times \frac{7-4\sqrt{7}+4}{3}=a\sqrt{7}+b
The square of \sqrt{7} is 7.
36If\times \frac{11-4\sqrt{7}}{3}=a\sqrt{7}+b
Add 7 and 4 to get 11.
12\left(11-4\sqrt{7}\right)If=a\sqrt{7}+b
Cancel out 3, the greatest common factor in 36 and 3.
\left(132-48\sqrt{7}\right)If=a\sqrt{7}+b
Use the distributive property to multiply 12 by 11-4\sqrt{7}.
\left(132I-48\sqrt{7}I\right)f=a\sqrt{7}+b
Use the distributive property to multiply 132-48\sqrt{7} by I.
132If-48\sqrt{7}If=a\sqrt{7}+b
Use the distributive property to multiply 132I-48\sqrt{7}I by f.
\left(132f-48\sqrt{7}f\right)I=a\sqrt{7}+b
Combine all terms containing I.
\left(-48\sqrt{7}f+132f\right)I=\sqrt{7}a+b
The equation is in standard form.
\frac{\left(-48\sqrt{7}f+132f\right)I}{-48\sqrt{7}f+132f}=\frac{\sqrt{7}a+b}{-48\sqrt{7}f+132f}
Divide both sides by 132f-48\sqrt{7}f.
I=\frac{\sqrt{7}a+b}{-48\sqrt{7}f+132f}
Dividing by 132f-48\sqrt{7}f undoes the multiplication by 132f-48\sqrt{7}f.
I=\frac{\left(4\sqrt{7}+11\right)\left(\sqrt{7}a+b\right)}{108f}
Divide a\sqrt{7}+b by 132f-48\sqrt{7}f.
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
Rationalize the denominator of \frac{\sqrt{7}-2}{\sqrt{7}+2} by multiplying numerator and denominator by \sqrt{7}-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
Consider \left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right). Multiplication can be transformed into difference of squares using the rule: \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)}{7-4}=a\sqrt{7}+b
Square \sqrt{7}. Square 2.
36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{3}=a\sqrt{7}+b
Subtract 4 from 7 to get 3.
36If\times \frac{\left(\sqrt{7}-2\right)^{2}}{3}=a\sqrt{7}+b
Multiply \sqrt{7}-2 and \sqrt{7}-2 to get \left(\sqrt{7}-2\right)^{2}.
36If\times \frac{\left(\sqrt{7}\right)^{2}-4\sqrt{7}+4}{3}=a\sqrt{7}+b
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{7}-2\right)^{2}.
36If\times \frac{7-4\sqrt{7}+4}{3}=a\sqrt{7}+b
The square of \sqrt{7} is 7.
36If\times \frac{11-4\sqrt{7}}{3}=a\sqrt{7}+b
Add 7 and 4 to get 11.
12\left(11-4\sqrt{7}\right)If=a\sqrt{7}+b
Cancel out 3, the greatest common factor in 36 and 3.
\left(132-48\sqrt{7}\right)If=a\sqrt{7}+b
Use the distributive property to multiply 12 by 11-4\sqrt{7}.
\left(132I-48\sqrt{7}I\right)f=a\sqrt{7}+b
Use the distributive property to multiply 132-48\sqrt{7} by I.
132If-48\sqrt{7}If=a\sqrt{7}+b
Use the distributive property to multiply 132I-48\sqrt{7}I by f.
\left(132f-48\sqrt{7}f\right)I=a\sqrt{7}+b
Combine all terms containing I.
\left(-48\sqrt{7}f+132f\right)I=\sqrt{7}a+b
The equation is in standard form.
\frac{\left(-48\sqrt{7}f+132f\right)I}{-48\sqrt{7}f+132f}=\frac{\sqrt{7}a+b}{-48\sqrt{7}f+132f}
Divide both sides by 132f-48\sqrt{7}f.
I=\frac{\sqrt{7}a+b}{-48\sqrt{7}f+132f}
Dividing by 132f-48\sqrt{7}f undoes the multiplication by 132f-48\sqrt{7}f.
I=\frac{\left(4\sqrt{7}+11\right)\left(\sqrt{7}a+b\right)}{108f}
Divide a\sqrt{7}+b by 132f-48\sqrt{7}f.
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
Rationalize the denominator of \frac{\sqrt{7}-2}{\sqrt{7}+2} by multiplying numerator and denominator by \sqrt{7}-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
Consider \left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right). Multiplication can be transformed into difference of squares using the rule: \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)}{7-4}=a\sqrt{7}+b
Square \sqrt{7}. Square 2.
36If\times \frac{\left(\sqrt{7}-2\right)\left(\sqrt{7}-2\right)}{3}=a\sqrt{7}+b
Subtract 4 from 7 to get 3.
36If\times \frac{\left(\sqrt{7}-2\right)^{2}}{3}=a\sqrt{7}+b
Multiply \sqrt{7}-2 and \sqrt{7}-2 to get \left(\sqrt{7}-2\right)^{2}.
36If\times \frac{\left(\sqrt{7}\right)^{2}-4\sqrt{7}+4}{3}=a\sqrt{7}+b
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{7}-2\right)^{2}.
36If\times \frac{7-4\sqrt{7}+4}{3}=a\sqrt{7}+b
The square of \sqrt{7} is 7.
36If\times \frac{11-4\sqrt{7}}{3}=a\sqrt{7}+b
Add 7 and 4 to get 11.
12\left(11-4\sqrt{7}\right)If=a\sqrt{7}+b
Cancel out 3, the greatest common factor in 36 and 3.
\left(132-48\sqrt{7}\right)If=a\sqrt{7}+b
Use the distributive property to multiply 12 by 11-4\sqrt{7}.
\left(132I-48\sqrt{7}I\right)f=a\sqrt{7}+b
Use the distributive property to multiply 132-48\sqrt{7} by I.
132If-48\sqrt{7}If=a\sqrt{7}+b
Use the distributive property to multiply 132I-48\sqrt{7}I by f.
a\sqrt{7}+b=132If-48\sqrt{7}If
Swap sides so that all variable terms are on the left hand side.
a\sqrt{7}=132If-48\sqrt{7}If-b
Subtract b from both sides.
\sqrt{7}a=-48\sqrt{7}If+132If-b
The equation is in standard form.
\frac{\sqrt{7}a}{\sqrt{7}}=\frac{-48\sqrt{7}If+132If-b}{\sqrt{7}}
Divide both sides by \sqrt{7}.
a=\frac{-48\sqrt{7}If+132If-b}{\sqrt{7}}
Dividing by \sqrt{7} undoes the multiplication by \sqrt{7}.
a=\frac{\sqrt{7}\left(-48\sqrt{7}If+132If-b\right)}{7}
Divide -b+132fI-48\sqrt{7}fI by \sqrt{7}.
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