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Componendo and Dividendo

Using propert...

Question

Using properties of proportion, solve for x :

(i) $\frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} + \sqrt{x - 16}} = \frac{7}{3}$

(ii) $\frac{\sqrt{x + 1} + \sqrt{x - 1}}{\sqrt{x + 1 - \sqrt{x - 1}}} = \frac{4 x - 1}{2}$

(iii) $\frac{3 x + \sqrt{9 x^{2} - 5}}{3 x + \sqrt{9 x^{2} - 5}} = 5$

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Solution

$left parenthesis i right parenthesis fraction numerator square root of x plus 5 end root plus square root of x minus 16 end root over denominator square root of x plus 5 end root minus square root of x minus 16 end root end fraction equals 7 over 3 A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o comma fraction numerator square root of x plus 5 end root plus square root of x minus 16 end root plus square root of x plus 5 end root minus square root of x minus 16 end root over denominator square root of x plus 5 end root plus square root of x minus 16 end root minus square root of x plus 5 end root plus square root of x minus 16 end root end fraction equals fraction numerator 7 plus 3 over denominator 7 minus 3 end fraction fraction numerator 2 square root of x plus 5 end root over denominator 2 square root of x minus 16 end root end fraction equals 10 over 4 fraction numerator square root of x plus 5 end root over denominator square root of x minus 16 end root end fraction equals 5 over 2 S q u a r i n g space b o t h space s i d e space comma fraction numerator x plus 5 over denominator x minus 16 end fraction equals 25 over 4 4 x plus 20 equals 25 x minus 400 21 x equals 420 x equals 420 over 21 equals 20 left parenthesis i i right parenthesis fraction numerator square root of x plus 1 end root plus square root of x minus 1 end root over denominator square root of x plus 1 end root minus square root of x minus 1 end root end fraction equals fraction numerator 4 x minus 1 over denominator 2 end fraction A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o comma fraction numerator square root of x plus 1 end root plus square root of x minus 1 end root plus square root of x plus 1 end root minus square root of x minus 1 end root over denominator square root of x plus 1 end root plus square root of x minus 1 end root minus square root of x plus 1 end root plus square root of x minus 1 end root end fraction equals fraction numerator 4 x minus 1 plus 2 over denominator 4 x minus 1 minus 2 end fraction fraction numerator 2 square root of x plus 1 end root over denominator 2 square root of x minus 1 end root end fraction equals fraction numerator 4 x plus 1 over denominator 4 x minus 3 end fraction S q u a r i n g space b o t h space s i d e s comma fraction numerator x plus 1 over denominator x minus 1 end fraction equals fraction numerator 16 x squared plus 1 plus 8 x over denominator 16 x squared plus 9 x minus 24 x end fraction A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o comma fraction numerator x plus 1 plus x minus 1 over denominator x plus 1 minus x plus 1 end fraction equals fraction numerator 16 x squared plus 1 plus 8 x plus 16 x squared plus 9 minus 24 x over denominator 16 x squared plus 1 plus 8 x minus 16 x squared minus 9 plus 24 x end fraction fraction numerator 2 x over denominator x end fraction equals fraction numerator 32 x squared plus 10 minus 16 x over denominator 32 x minus 8 end fraction x equals fraction numerator 16 x squared plus 5 minus 8 x over denominator 16 x minus 4 end fraction 16 x squared minus 4 x equals 16 x squared plus 5 minus 8 x 4 x equals 5 x equals 5 over 4 left parenthesis i i i right parenthesis fraction numerator 3 x plus square root of 9 x squared minus 5 end root over denominator 3 x minus square root of 9 x squared minus 5 end root end fraction equals 5 A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o fraction numerator 3 x plus square root of 9 x squared minus 5 end root plus 3 x minus square root of 9 x squared minus 5 end root over denominator 3 x plus square root of 9 x squared minus 5 end root minus 3 x plus square root of 9 x squared minus 5 end root end fraction equals fraction numerator 5 plus 1 over denominator 5 minus 1 end fraction fraction numerator 6 x over denominator 2 square root of 9 x squared minus 5 end root end fraction equals 6 over 4 fraction numerator x over denominator square root of 9 x squared minus 5 end root end fraction equals 1 half S q u a r i n g space b o t h space s i d e s comma fraction numerator x squared over denominator 9 x squared minus 5 end fraction equals 1 fourth 4 x squared equals 9 x squared minus 5 5 x squared equals 5 x squared equals 1 x equals plus-or-minus 1$

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Q. Solve for x:

(i)

\frac{x - 1}{x - 2} + \frac{x - 3}{x - 4} = 3 \frac{1}{3}; x \neq 2, 4

\frac{1}{x} - \frac{1}{x - 2} = 3, x \neq 0, 2

x + \frac{1}{x} = 3, x \neq 0

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Q. Using properties of proportion, solve for

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i. \frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{7}{3}

Q. Using properties of proportion, solve for

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\frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{7}{3}

Q. Using properties of proportion solve:

\frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{7}{3}

Q. Using properties of proportion, solve for x:

(i i i) \frac{3 x + \sqrt{9 x^{2} - 5}}{3 x - \sqrt{9 x^{2} - 5}} = 5

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