Answer:
k + 2 = 3√5
Explanation:
[tex]\sf Given: k = \sqrt{49 - 12\sqrt{5} }[/tex]
[tex]\sf \rightarrow k = \sqrt{49 - 12\sqrt{5} }[/tex]
step 1: breakdown
[tex]\sf \rightarrow k = \sqrt{4 + 45- 2 \times 2 \times 3 \sqrt{5} }[/tex]
step 2: rewrite 45 = (3√5)²
[tex]\sf \rightarrow k = \sqrt{4 + (3\sqrt{5})^2 - 2 \times 2 \times 3 \sqrt{5} }[/tex]
step 3: rearrange as per a² -2ab + b² = (a - b)²
[tex]\sf \rightarrow k = \sqrt{ (3\sqrt{5})^2 - 2 \times 3 \sqrt{5} \times 2 +2^2}[/tex]
step 4: comparing, here: a = 3√5, b = 2
[tex]\sf \rightarrow k = \sqrt{ (3\sqrt{5} -2)^2}[/tex]
step 5: simplify √a² = a
[tex]\sf \rightarrow k = 3\sqrt{5} -2[/tex]
Then find k + 2
[tex]\rightarrow \sf k = 3\sqrt{5} -2 + 2[/tex]
[tex]\rightarrow \sf k = 3\sqrt{5}[/tex]
