Problem 4 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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, \( \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6} \).

Step 2: Substitute and Simplify

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

Step 3: Split the Fraction into Two Parts

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

Step 4: 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.

Step 5: Combine Simplified Expressions

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

Key Concepts

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.