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Chapter 10: Problem 47

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this. $$ \sqrt{4 x^{2}+28 x+49} $$

Short Answer

Expert verified
|2x + 7|

Step by step solution

01

Identify Polynomial

Identify the polynomial under the square root: \[ \ 4x^2 + 28x + 49 \]
02

Rewrite Polynomial

Rewrite the polynomial in a form that may indicate a perfect square trinomial: \[ \ 4x^2 + 28x + 49 = (2x)^2 + 2 \times 2x \times 7 + 7^2 \]
03

Express as a Square

Recognize that \[ 4x^2 + 28x + 49 = (2x + 7)^2 \]
04

Take the Square Root

Simplify the square root of both sides: \[ \ \sqrt{4x^2 + 28x + 49} = \sqrt{(2x + 7)^2} \]
05

Apply Absolute Value

The square root of a square encompasses absolute value: \[ \ \sqrt{(2x + 7)^2} = |2x + 7| \]

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Key Concepts

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

absolute value
The absolute value of a number or expression measures its distance from zero on a number line. This value is always non-negative, irrespective of the original number being positive or negative.
Consider an example: \, |x|. If \(x\) is 3, \(|x| = 3\). Likewise, if \(x\) is -3, then \(|x| = 3\) as well.
Absolute value becomes particularly significant when dealing with expressions under a square root. For example, simplifying \(\sqrt{\left(2x + 7\right)^2}\) results in \(|2x + 7|\). Here, the absolute value ensures the expression remains non-negative.
Remember, when taking the square root of a square, always use absolute value notation: \(\sqrt{a^2} = |a|\).
perfect square trinomials
A perfect square trinomial is a quadratic expression of the form \(a^2 + 2ab + b^2\), which can be factored into \((a + b)^2\).
Recognizing patterns in the trinomial helps in simplification:
  • Identify terms that are squares: \(4x^2 + 28x + 49\)
  • Rewrite as: \((2x)^2 + 2(2x)(7) + 7^2\)
  • Recognize this matches the form: \((a+b)^2\)
  • Which simplifies to: \((2x + 7)^2\)

Perfect square trinomials simplify computation involving square roots by organizing complex expressions into more manageable forms.
Always look out for squared terms, cross-products, and constant squares in quadratic expressions to identify perfect square trinomials.
polynomial simplification
Polynomial simplification is the process of reducing a polynomial to its simplest form. This involves combining like terms and factoring where possible.
Steps to simplify a polynomial:
  • Identify all like terms鈥攖erms with the same variable raised to the same power.
  • Combine like terms.
  • Factor the polynomial where possible.
For the expression \(4x^2 + 28x + 49\), we can:
  • Identify that it's a perfect square: \((2x + 7)^2\)
  • Thus, simplifying it directly to \((2x + 7)^2\)
Polynomial simplification makes solving equations and finding roots easier.
It transforms complex expressions into understandable and manageable forms.
square root
The square root function, denoted by \(\sqrt{...}\), finds a number that, when multiplied by itself, gives the original number.
For instance, \(\sqrt{9} = 3\) and \(\sqrt{16} = 4\).
When working with expressions under a square root, recognizing perfect square patterns is crucial.
For the expression \(\sqrt{(2x + 7)^2}\):
  • We identify it as a perfect square.
  • Taking the square root simplifies it: \(|2x + 7|\).
This procedure underscores the need for identifying perfect squares within expressions.
This allows simpler and clearer solutions in algebra and calculus, promoting better mathematical understanding.

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Most popular questions from this chapter

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{s^{2} t^{4}} \sqrt[3]{s^{4} t^{6}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt{(7-y)^{3}}}{\sqrt[3]{(7-y)^{2}}}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\sqrt[3]{x y^{2} z} \sqrt{x^{3} y z^{2}}$$

The absolute value of a complex number \(a+b i\) is its distance from the origin. Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}} .\) Find the absolute value of each complex number. $$ |8-6 i| $$

Multiply. $$(\sqrt{x+2}-\sqrt{x-2})^{2}$$
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