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Simplify. Rationalize the denominator. \newline4−5+2\frac{4}{-5 + \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline4−5+2\frac{4}{-5 + \sqrt{2}}
  1. Identify Conjugate: Select the conjugate of −5+2-5 + \sqrt{2}.\newlineThe conjugate of a number of the form a+ba + \sqrt{b} is a−ba - \sqrt{b}. Therefore, the conjugate of −5+2-5 + \sqrt{2} is −5−2-5 - \sqrt{2}.
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator.\newline(4⋅(−5−2))/((−5+2)⋅(−5−2))(4 \cdot (-5 - \sqrt{2}))/((-5 + \sqrt{2}) \cdot (-5 - \sqrt{2}))
  3. Simplify Numerator: Simplify the numerator.\newlineNow we distribute the 44 across the terms in the conjugate.\newline4×(−5)+4×(−2)4 \times (-5) + 4 \times (-\sqrt{2})\newline=−20−42= -20 - 4\sqrt{2}
  4. Simplify Denominator: Simplify the denominator.\newlineWe use the difference of squares formula, which states that (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.\newline(−5)2−(2)2(-5)^2 - (\sqrt{2})^2\newline= 2525 - 22\newline= 2323
  5. Write Simplified Expression: Write the simplified expression.\newlineNow we have the simplified numerator and denominator.\newline(−20−42)/23(-20 - 4\sqrt{2}) / 23\newlineThis fraction is already in simplest form.

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