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Chapter 6: Problem 4

Simplify and reduce each expression. $$ \frac{-9 \pm \sqrt{54}}{3} $$

Short Answer

Expert verified
Final result: \( -3 \pm \sqrt{6} \).

Step by step solution

01

Simplify the Square Root

First, simplify the square root of 54. We know that 54 can be factored into 9 and 6, where 9 is a perfect square. Thus, \( \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6} \).
02

Substitute and Simplify

Substitute \( \sqrt{54} \) with \( 3\sqrt{6} \) in the expression. Now we have: \[ \frac{-9 \pm 3\sqrt{6}}{3} \].
03

Split the Fraction into Two Parts

Split the expression \( \frac{-9 \pm 3\sqrt{6}}{3} \) into two separate fractions: \( \frac{-9}{3} \pm \frac{3\sqrt{6}}{3} \).
04

Simplify Each Fraction

Simplify each fraction separately: - \( \frac{-9}{3} = -3 \).- \( \frac{3\sqrt{6}}{3} = \sqrt{6} \), because the 3's cancel each other out.
05

Combine Simplified Expressions

Combine the simplified terms: \[ -3 \pm \sqrt{6} \]. This is the simplified form of the original expression.

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Key Concepts

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

Square Roots
A square root is essentially the opposite of squaring a number. For example, if we square 3, we get 9. Then, the square root of 9 is 3. In mathematical terms, the square root of a number, say \( x \), is a value that, when multiplied by itself, gives \( x \).
To simplify a square root, you look for perfect square factors of the number under the root. A perfect square is any number that is the square of an integer.
  • In our exercise, we simplified \( \sqrt{54} \).
  • Since 54 is equal to \( 9 \times 6 \), and 9 is a perfect square, it simplifies to \( 3\sqrt{6} \).
When you identify and separate perfect squares, it becomes easier to take them out of the square root.
Factoring
Factoring is simply breaking down a number or an expression into its component parts or 'factors'. These are numbers or expressions you can multiply together to get the original number or expression. It's a crucial concept in algebra that helps simplify various expressions.
Factoring often involves techniques such as:
  • Finding greatest common factors
  • Factoring trinomials
  • Decomposing expressions into products of simpler expressions
In our case, 54 was factored into 9 and 6. This simplification allowed us to easily handle the square root.
Quadratic Formula
The quadratic formula is a steadfast method used to solve quadratic equations. A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \). The quadratic formula comes to the rescue, providing solutions at a glance. Here it is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The elements inside the formula:
  • \( a \), \( b \), and \( c \) are numbers from your quadratic equation.
  • The symbol \( \pm \) indicates that the formula gives two possible solutions.
The expression from the exercise is reminiscent of this form. It helps to remember such patterns when dealing with quadratic problems.
Fractions
Fractions involve numbers in the form of \( \frac{numerator}{denominator} \). They often require simplification for cleaner, more manageable expressions. Simplifying fractions involves reducing them to their simplest form.
  • This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In the expression \( \frac{-9 \pm 3\sqrt{6}}{3} \), splitting into two fractions simplifies each part.
  • \( \frac{-9}{3} = -3 \) simplifies beautifully.
  • \( \frac{3\sqrt{6}}{3} \) simplifies to \( \sqrt{6} \), with 3s canceling out.
Understanding fractions is essential as it makes algebraic expressions and calculations easier to manage.

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Most popular questions from this chapter

Solve each inequality. $$ 9 x^{2}-6 x+1 \leq 0 $$

Set up an equation and solve each problem. A rectangular piece of cardboard is 2 units longer than it is wide. From each of its corners a square piece 2 units on a side is cut out. The flaps are then turned up to form an open box that has a volume of 70 cubic units. Find the length and width of the original piece of cardboard.

Solve each inequality. $$ \frac{x}{3 x+7} \geq 0 $$

Solve each inequality and graph its solution set on a number line. $$ (x+1)(x-1)(x-3)>0 $$

The product \((x-2)(x+3)\) is positive if both factors are negative or if both factors are positive. Therefore, we can solve \((x-2)(x+3)>0\) as follows: \((x-2<0\) and \(x+3<0)\) or \((x-2>0\) and \(x+3>0)\) \((x<2\) and \(x<-3)\) or \((x>2\) and \(x>-3)\) $$ x<-3 \text { or } x>2 $$ The solution set is \((-\infty,-3) \cup(2, \infty)\). Use this type of analysis to solve each of the following. (a) \((x-2)(x+7)>0\) (b) \((x-3)(x+9) \geq 0\) (c) \((x+1)(x-6) \leq 0\) (d) \((x+4)(x-8)<0\) (e) \(\frac{x+4}{x-7}>0\) (f) \(\frac{x-5}{x+8} \leq 0\)
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