If \frac{2\sqrt{3} - \sqrt{5}}{\sqrt{3} + \sqrt{5}} = a + b\sqrt{15}, where a and b are ra… — JAMB UTME

If $\frac{2\sqrt{3} - \sqrt{5}}{\sqrt{3} + \sqrt{5}} = a + b\sqrt{15}$ , where $a$ and $b$ are rational numbers, what is the value of $a - b$ ?

-7

Rationalizing

\frac{2\sqrt{3} - \sqrt{5}}{\sqrt{3} + \sqrt{5}}

by multiplying top and bottom by

(\sqrt{3} - \sqrt{5})

yields

\frac{11 - 3\sqrt{15}}{-2} = -\frac{11}{2} + \frac{3}{2}\sqrt{15}

. Comparing this with

a + b\sqrt{15}

gives

a = -\frac{11}{2}

and

b = \frac{3}{2}

. Computing

a - b

gives

-\frac{11}{2} - \frac{3}{2} = -7

Multiply the numerator and denominator by the conjugate of the denominator

(\sqrt{3} - \sqrt{5})

\frac{(2\sqrt{3} - \sqrt{5})(\sqrt{3} - \sqrt{5})}{(\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5})}

Rationalizing the denominator requires using the difference of squares.

Expand the numerator and simplify the denominator

Numerator:

2(3) - 2\sqrt{15} - \sqrt{15} + 5 = 11 - 3\sqrt{15}

. Denominator:

3 - 5 = -2

Multiply terms using FOIL and replace

(\sqrt{3})^2

with

3

and

(\sqrt{5})^2

with

5

Divide the numerator by the denominator to express in standard form

a + b\sqrt{15}

\frac{11 - 3\sqrt{15}}{-2} = -\frac{11}{2} + \frac{3}{2}\sqrt{15}

Separate the rational term and the surd coefficient.

Equate coefficients to find

a

and

b

, then compute

a - b

a = -\frac{11}{2}, b = \frac{3}{2} \implies a - b = -\frac{11}{2} - \frac{3}{2} = -\frac{14}{2} = -7

Calculate the target expression

a - b

using the derived rational values.

Rationalization of Binomial Denominators and Equating Surd Coefficients

Question