If x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} and y = \frac{\sqrt{7} - \sqrt{3}}{… — JAMB UTME

If $x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}$ and $y = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}$ , determine the numerical value of $x^2 + y^2$ .

Cevap: 23

Cevap

The numerical value of

x^2 + y^2

23

Rationalizing

x

yields

\frac{5 + \sqrt{21}}{2}

and rationalizing

y

yields

\frac{5 - \sqrt{21}}{2}

. The sum

x + y

equals

5

and the product

xy

equals

1

. Substituting these into

x^2 + y^2 = (x + y)^2 - 2xy

gives

5^2 - 2(1) = 23

Adım Adım Çözüm

Rationalize the denominators of

x

and

y

x = \frac{5 + \sqrt{21}}{2}

and

y = \frac{5 - \sqrt{21}}{2}

Multiply the numerator and denominator by the conjugate of the denominator.

Calculate the sum

x + y

and the product

xy

x + y = 5

and

xy = 1

Summing conjugate surd expressions cancels the radical term, and multiplying them applies the difference of two squares.

Evaluate

x^2 + y^2

using the identity

(x + y)^2 - 2xy

x^2 + y^2 = 5^2 - 2(1) = 23

Substituting the known sum and product avoids having to square complex surd expressions directly.

Anahtar Kavram

Rationalization of binomial denominators and application of symmetric algebraic identities.

Soru

Cevap

Adım Adım Çözüm

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