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Chapter 7: Problem 46

Find a simplified form of \(f(x) .\) Assume that \(x\) can be any real number. $$f(x)=\sqrt{2 x^{2}+8 x+8}$$

Short Answer

Expert verified
Simplified form is.

Step by step solution

01

- Recognize the Quadratic Expression

The expression inside the square root is a quadratic expression in the form of ax^2 + bx + c . Identify the coefficients: a = 2, b = 8, c = 8.
02

- Find the Perfect Square Form

Rewrite the quadratic expression 2x^2 + 8x + 8 as a perfect square. We factor out the 2: 2(x^2 + 4x) + 8 . Now complete the square for x^2 + 4x . Take half of 4, square it to get 4, and add and subtract it inside the parenthesis: 2(x^2 + 4x + 4 - 4) + 8 . This simplifies to 2((x + 2)^2 - 4) + 8 .
03

- Simplify Inside the Square Root

Distribute the 2 back into the expression inside the square root: 2((x + 2)^2 - 4) + 8 = 2(x + 2)^2 - 8 + 8 = 2(x + 2)^2 . Now the expression inside the square root is f(x) = (sqrt( 2 (x+2)^2 ))
04

- Apply the Square Root to the Simplified Expression

Take the square root of the simplified expression: f(x) = sqrt{2 * (x+2)^2} = sqrt(2) * sqrt((x+2)^2) sqrt((x+2)^2) = 2
05

- Write the Final Simplified Form

The final simplified form

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Expression
When working with quadratic expressions, you're dealing with a polynomial of degree two. This typically looks like \(ax^2 + bx + c\). Identifying the coefficients \(a\), \(b\), and \(c\) is the first step. In our exercise:
- \(a = 2\)
- \(b = 8\)
- \(c = 8\).
Recognizing these components helps you understand the structure of the expression and plan the steps needed to simplify it.
Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This is useful for solving quadratic equations and simplifying expressions. For our quadratic \(2x^2 + 8x + 8\):
- Factor out the 2 from \(2x^2 + 8x\): \(2(x^2 + 4x) + 8\)
- To complete the square inside the parenthesis, take half of the coefficient of \(x\), square it, and add and subtract it: \(2(x^2 + 4x + 4 - 4) + 8\)
- This becomes \(2((x + 2)^2 - 4) + 8\).
This step makes it easier to work with the expression under the square root by making it a perfect square trinomial.
Square Root Property
The square root property allows us to simplify expressions involving square roots, especially when dealing with perfect squares. After completing the square:
- Rewrite the expression: \(2((x + 2)^2 - 4) + 8 = 2(x + 2)^2 - 8 + 8 = 2(x + 2)^2\)
- Now, apply the square root to both sides: \(\sqrt{2(x + 2)^2}\)
- Employ the property \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\): \(\sqrt{2} \cdot \sqrt{(x + 2)^2}\).
- This simplifies further as \(\sqrt{(x + 2)^2} = |x + 2|\). Therefore, we get: \(\sqrt{2} |x + 2|\). Understanding this property is fundamental in simplifying quadratic expressions.
Factoring
Factoring is a technique used to rewrite expressions as a product of simpler expressions. It plays a key role in solving and simplifying quadratic equations. In our problem, we factor to find the perfect square form:
- Start with \(2(x^2 + 4x)\)
- Complete the square to get \(2((x + 2)^2 - 4)\)
- Simplify further to \(2(x + 2)^2 - 8 + 8\)
- Finally, we have \(2(x + 2)^2\).
Factoring not only simplifies the quadratic expression but also makes it easier to apply other properties like the square root property.

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