Question

quiz instructions

for this quiz answer each question.
you have 20 minutes to complete the quiz.
you have only one attempt.

question 4

simplify the expression.

\\((-2 + \sqrt{3})(-7 + \sqrt{3})\\)

\\(\bigcirc\\ \sqrt{105} + 14\sqrt{2}\\)
\\(\bigcirc\\ 17\\)
\\(\bigcirc\\ 17 - 9\sqrt{3}\\)
\\(\bigcirc\\ 5 + \sqrt{15} - 8\sqrt{30} - 24\sqrt{2}\\)

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Expand the expression

Multiply the binomials using the FOIL method (First, Outer, Inner, Last):

$$ (-2 + \sqrt{3})(-7 + \sqrt{3}) = (-2)(-7) + (-2)(\sqrt{3}) + (\sqrt{3})(-7) + (\sqrt{3})(\sqrt{3}) $$

Step 2: Simplify each term

Calculate the product of each term:

$$ (-2)(-7) = 14 $$

$$ (-2)(\sqrt{3}) = -2\sqrt{3} $$

$$ (\sqrt{3})(-7) = -7\sqrt{3} $$

$$ (\sqrt{3})(\sqrt{3}) = 3 $$

Combine these simplified terms:

$$ 14 - 2\sqrt{3} - 7\sqrt{3} + 3 $$

Step 3: Combine like terms

Group the rational numbers and the radical terms together:

$$ (14 + 3) + (-2\sqrt{3} - 7\sqrt{3}) $$

$$ 17 - 9\sqrt{3} $$

Answer:

\(17 - 9\sqrt{3}\)