In Exercises 33–38, express the function, f, in simplified form. Assume that x can be any real number.___________f(x) = √5x² - 10x + 5

Verified step by step guidance

1. Start by identifying the expression inside the square root: \(5x^2 - 10x + 5\).
2. Factor the quadratic expression inside the square root. Look for a common factor in all terms. In this case, factor out 5: \(5(x^2 - 2x + 1)\).
3. Notice that the expression inside the parentheses, \(x^2 - 2x + 1\), is a perfect square trinomial. It can be rewritten as \((x - 1)^2\).
4. Substitute the factored expression back into the function: \(f(x) = \sqrt{5((x - 1)^2)}\).
5. Simplify the expression by taking the square root of the perfect square: \(f(x) = \sqrt{5} \cdot |x - 1|\). This is the simplified form of the function.>

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. In this case, the expression under the square root, 5x² - 10x + 5, is a quadratic function. Understanding its properties, such as the vertex, axis of symmetry, and roots, is essential for simplifying the function.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique allows for easier simplification and analysis of the function. By rewriting the quadratic in the form (x - p)² = q, we can simplify the expression under the square root and facilitate further calculations.

Square Roots and Simplification

The square root function, denoted as √, is used to find a number that, when multiplied by itself, gives the original number. Simplifying expressions involving square roots often requires factoring out perfect squares or recognizing patterns. In this problem, simplifying √(5x² - 10x + 5) involves identifying and extracting any perfect square factors to express the function in a more manageable form.