Question

$simplify$
$(3 + 4 \sqrt{5})(2 - x \sqrt{5}) = 2 \sqrt{5} - 34$
​

Answer 1

Answer:

x = 2

Step-by-step explanation:

Given

$(3+4\sqrt{5})(2-x\sqrt{5})=2\sqrt{5}-34$

First, open the brackets using distributive property:

$(3+4\sqrt{5})(2-x\sqrt{5})\\ \\=3(2-x\sqrt{5})+4\sqrt{5}(2-x\sqrt{5})\\ \\=6-3x\sqrt{5}+8\sqrt{5}-4\sqrt{5}x\sqrt{5}\\ \\=6-3\sqrt{5}x+8\sqrt{5}-20x\ [\text{since} \sqrt{5}\cdot \sqrt{5}=5, \ \text{then }4\sqrt{5}\cdot \sqrt{5}=20]$

Now, combine the like terms:

$6-3\sqrt{5}x+8\sqrt{5}-20x\\ \\(6-20x)+\sqrt{5}(8-3x)$

Since $(3+4\sqrt{5})(2-x\sqrt{5})=2\sqrt{5}-34,$ you have that

$6-20x=-34\Rightarrow 20x=40,\ x=2\\ \\8-3x=2\Rightarrow 3x=6,\ x=2$