Find the value of ' a ' in (3-sqrt5)/(3+2sqrt5)=asqrt5-(19)/(11)

Rationalize Denominator: First, let's rationalize the denominator of the left side of the equation by multiplying both the numerator and denominator by the conjugate of the denominator. (3-sqrt5)/(3+2sqrt5) * (3-2sqrt5)/(3-2sqrt5)Multiply Numerators and Denominators: Now, multiply the numerators and the denominators. Numerator: (3-sqrt5)(3-2sqrt5) Denominator: (3+2sqrt5)(3-2sqrt5)Expand Using FOIL Method: Use the FOIL method to expand the numerator and denominator. Numerator: 3*3 - 3*2sqrt5 - sqrt5*3 + sqrt5*2sqrt5 Denominator: 3*3 - 3*2sqrt5 + 2sqrt5*3 - 2sqrt5*2sqrt5Simplify Expressions: Simplify the expressions. Numerator: 9 - 6sqrt5 - 3sqrt5 + 2*5 Denominator: 9 - 6sqrt5 + 6sqrt5 - 2*5Combine Like Terms: Combine like terms. Numerator: 9 + 10 - 9sqrt5 Denominator: 9 - 10Further Simplify Terms: Simplify the terms further. Numerator: 19 - 9sqrt5 Denominator: -1Divide Numerator by Denominator: Divide the numerator by the denominator. (19 - 9sqrt5) / -1 = -19 + 9sqrt5Set Equal to Right Side: Now we have the left side of the equation simplified to -19 + 9sqrt5. Set this equal to the right side of the original equation. -19 + 9sqrt5 = asqrt5 - 19/11Find 'a': To find 'a', we need to compare the coefficients of sqrt5 on both sides of the equation. 9 = a

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Find the value of '
a ' in
(3-sqrt5)/(3+2sqrt5)=asqrt5-(19)/(11)

Find the value of ' $a$ ' in $\frac{3-\sqrt{5}}{3+2 \sqrt{5}}=a \sqrt{5}-\frac{19}{11}$

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Algebra 2

Composition of linear functions: find a value

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Q. Find the value of '

a

' in

\frac{3-\sqrt{5}}{3+2 \sqrt{5}}=a \sqrt{5}-\frac{19}{11}

Rationalize Denominator: First, let's rationalize the denominator of the left side of the equation by multiplying both the numerator and denominator by the conjugate of the denominator. $\newline$ $(3-\sqrt{5})/(3+2\sqrt{5}) \times (3-2\sqrt{5})/(3-2\sqrt{5})$
Multiply Numerators and Denominators: Now, multiply the numerators and the denominators. $\newline$ Numerator: $(3-\sqrt{5})(3-2\sqrt{5})$ $\newline$ Denominator: $(3+2\sqrt{5})(3-2\sqrt{5})$
Expand Using FOIL Method: Use the FOIL method to expand the numerator and denominator. $\newline$ Numerator: $3\times3 - 3\times2\sqrt{5} - \sqrt{5}\times3 + \sqrt{5}\times2\sqrt{5}$ $\newline$ Denominator: $3\times3 - 3\times2\sqrt{5} + 2\sqrt{5}\times3 - 2\sqrt{5}\times2\sqrt{5}$
Simplify Expressions: Simplify the expressions. $\newline$ Numerator: $9 - 6\sqrt{5} - 3\sqrt{5} + 2\cdot 5$ $\newline$ Denominator: $9 - 6\sqrt{5} + 6\sqrt{5} - 2\cdot 5$
Combine Like Terms: Combine like terms. $\newline$ Numerator: $9 + 10 - 9\sqrt{5}$ $\newline$ Denominator: $9 - 10$
Further Simplify Terms: Simplify the terms further. $\newline$ Numerator: $19 - 9\sqrt{5}$ $\newline$ Denominator: $-1$
Divide Numerator by Denominator: Divide the numerator by the denominator. $\newline$ $(19 - 9\sqrt{5}) / -1 = -19 + 9\sqrt{5}$
Set Equal to Right Side: Now we have the left side of the equation simplified to $-19 + 9\sqrt{5}$ . Set this equal to the right side of the original equation. $-19 + 9\sqrt{5} = a\sqrt{5} - \frac{19}{11}$
Find 'a': To find 'a', we need to compare the coefficients of $\sqrt{5}$ on both sides of the equation. $9 = a$