Simplify (3 + 4 √(5))(2 - x √(5)) = 2 √(5) - 34 ​

Question

`simplify`

`(3 + 4 √(5))(2 - x √(5)) = 2 √(5) - 34`
​

Answer 1

x = 2

Explanation:

Given

`(3+4√(5))(2-x√(5))=2√(5)-34`

First, open the brackets using distributive property:

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

Now, combine the like terms:

`6-3√(5)x+8√(5)-20x\\ \\(6-20x)+√(5)(8-3x)`

Since
`(3+4√(5))(2-x√(5))=2√(5)-34,` you have that

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