Question

`\sqrt[3]{4x^(2)-20x+25 }`

Answer 1

The expression
`\sqrt[3]{4x^(2) - 20x + 25}` represents the cube root of the quadratic trinomial
`4x^(2) - 20x + 25`. To understand this expression, let's break it down step by step.

Firstly, the quadratic trinomial inside the cube root,
`4x^(2) - 20x + 25`, can be factored into the perfect square
`(2x - 5)^(2)`. This factorization is obtained by recognizing that the trinomial has the form
`a^(2) - 2ab + b^(2)` , where a=2x and b=5. Thus,
`(2x - 5)^(2)` is equivalent to
`4x^(2) - 20x + 25`.

Now, the cube root of this perfect square,
`\sqrt[3]{(2x-5)^(2) }`​ , can be further simplified to
`2x-5`. This is because the cube root is the inverse operation of raising a number to the power of 3, and since
`(2x-5)^(3) = 4x^(2) - 20x + 25`, taking the cube root of
`(2x-5)^(2)` yields
`2x-5`.

In summary,
`\sqrt[3]{4x^(2) - 20x + 25}` simplifies to 2x−5. This expression is valuable in solving equations or expressing mathematical relationships where the cube root of a quadratic expression is involved. The process involves recognizing the perfect square within the trinomial and then applying the cube root to obtain the simplified form.

Complete question

Solve the given expression
`\sqrt[3]{4x^(2) - 20x + 25}`