Solve quadtext{36.If}frac{sqrt{7}-2}{sqrt{7}+2}=asqrt{7}+b

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

Solve for I (complex solution)

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

Steps for Solving Linear Equation

Rationalize the denominator of \frac{\sqrt{7}-2}{\sqrt{7}+2} by multiplying numerator and denominator by \sqrt{7}-2.

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

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}.

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

Square \sqrt{7}. Square 2.

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

Subtract 4 from 7 to get 3.

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

Multiply \sqrt{7}-2 and \sqrt{7}-2 to get \left(\sqrt{7}-2\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{7}-2\right)^{2}.

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

The square of \sqrt{7} is 7.

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

Add 7 and 4 to get 11.

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

Cancel out 3, the greatest common factor in 36 and 3.

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

Use the distributive property to multiply 12 by 11-4\sqrt{7}.

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

Use the distributive property to multiply 132-48\sqrt{7} by I.

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

Use the distributive property to multiply 132I-48\sqrt{7}I by f.

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

Combine all terms containing I.

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

The equation is in standard form.

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

Divide both sides by 132f-48\sqrt{7}f.

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

Dividing by 132f-48\sqrt{7}f undoes the multiplication by 132f-48\sqrt{7}f.

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

Divide a\sqrt{7}+b by 132f-48\sqrt{7}f.

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

Solve for I

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

Steps for Solving Linear Equation

Rationalize the denominator of \frac{\sqrt{7}-2}{\sqrt{7}+2} by multiplying numerator and denominator by \sqrt{7}-2.

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

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}.

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

Square \sqrt{7}. Square 2.

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

Subtract 4 from 7 to get 3.

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

Multiply \sqrt{7}-2 and \sqrt{7}-2 to get \left(\sqrt{7}-2\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{7}-2\right)^{2}.

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

The square of \sqrt{7} is 7.

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

Add 7 and 4 to get 11.

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

Cancel out 3, the greatest common factor in 36 and 3.

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

Use the distributive property to multiply 12 by 11-4\sqrt{7}.

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

Use the distributive property to multiply 132-48\sqrt{7} by I.

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

Use the distributive property to multiply 132I-48\sqrt{7}I by f.

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

Combine all terms containing I.

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

The equation is in standard form.

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

Divide both sides by 132f-48\sqrt{7}f.

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

Dividing by 132f-48\sqrt{7}f undoes the multiplication by 132f-48\sqrt{7}f.

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

Divide a\sqrt{7}+b by 132f-48\sqrt{7}f.

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

Solve for a

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

Steps for Solving Linear Equation

Rationalize the denominator of \frac{\sqrt{7}-2}{\sqrt{7}+2} by multiplying numerator and denominator by \sqrt{7}-2.

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

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}.

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

Square \sqrt{7}. Square 2.

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

Subtract 4 from 7 to get 3.

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

Multiply \sqrt{7}-2 and \sqrt{7}-2 to get \left(\sqrt{7}-2\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{7}-2\right)^{2}.

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

The square of \sqrt{7} is 7.

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

Add 7 and 4 to get 11.

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

Cancel out 3, the greatest common factor in 36 and 3.

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

Use the distributive property to multiply 12 by 11-4\sqrt{7}.

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

Use the distributive property to multiply 132-48\sqrt{7} by I.

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

Use the distributive property to multiply 132I-48\sqrt{7}I by f.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract b from both sides.

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

The equation is in standard form.

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

Divide both sides by \sqrt{7}.

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

Dividing by \sqrt{7} undoes the multiplication by \sqrt{7}.

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

Divide -b+132fI-48\sqrt{7}fI by \sqrt{7}.

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