Problem 19 - A Graphical Approach to College Algebra

Question

Answer each question. For what values of \(x\) is \(\sqrt{9 a x^{2}}=3 x \sqrt{a}\) a true statement? Assume that \(a \geq 0\)

Step-by-Step Solution

Answer

The equation is true for \( x \geq 0 \).

Step 1: Understand the equation

We need to solve the equation \( \sqrt{9ax^2} = 3x\sqrt{a} \) for \( x \). The equation involves square roots, and both sides should match for given \( x \) and \( a \).

Step 2: Simplify both sides

First, simplify the left side \( \sqrt{9ax^2} \). This can be rewritten as \( \sqrt{9} \cdot \sqrt{a} \cdot \sqrt{x^2} = 3\sqrt{a}|x| \) because \( \sqrt{x^2} = |x| \).

Step 3: Equate the simplified expressions

The equation becomes \( 3\sqrt{a}|x| = 3x\sqrt{a} \). Since \( 3\sqrt{a} eq 0 \), divide both sides by \( 3\sqrt{a} \) to get \( |x| = x \).

Step 4: Solve the absolute value equation

The equation \( |x| = x \) holds true when \( x \geq 0 \). For negative \( x \), \( |x| = -x \) which does not satisfy \( |x| = x \).

Step 5: Consider the constraints

Since \( a \geq 0 \) and the square root is defined, no additional constraints come from \( a \). Thus, the constraint \( x \geq 0 \) is sufficient.

Key Concepts

Square Roots in Algebraic Equations

Square roots play a crucial role when dealing with algebraic equations that involve finding values that satisfy certain conditions. In the exercise, we start with the expression

• \( \sqrt{9ax^2} \)

which we need to simplify.
This expression can be broken down since square roots can be separated. This gives us \( \sqrt{9} \cdot \sqrt{a} \cdot \sqrt{x^2} \). The square root of any positive number involves finding another number which, when multiplied by itself, gives the original number.
Here are simple breakdowns:

• \( \sqrt{9} = 3 \)
• \( \sqrt{x^2} = |x| \)

This results in the expression \( 3\sqrt{a}|x| \), transforming the original equation to better compare both sides.
Understanding how to manipulate and simplify square roots is foundational in solving and simplifying algebraic equations.

Understanding Absolute Value

The concept of absolute value is key when dealing with equations involving square roots and powers. Absolute value, denoted as \(|x|\), represents the distance of a number from zero on the number line.
It's important to note:

• If \( x \) is positive or zero, \(|x| = x\)
• If \( x \) is negative, \(|x| = -x\)

In the given equation \( |x| = x \), we find that this holds true when \( x \geq 0 \). This means the expression is true for any non-negative value of \( x \).
Absolute values are frequently encountered in algebraic manipulations, especially when solving equations to ensure all potential values of variables are considered.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involves rewriting them in a form that is easier to work with. In the original equation, simplifying is achieved by transforming expressions into more convenient forms.
We took the expression \( \sqrt{9ax^2} \) and broke it down into \( 3\sqrt{a}|x| \), allowing us to equate both sides directly:
\( 3\sqrt{a}|x| = 3x\sqrt{a} \).
Since \( 3\sqrt{a} eq 0 \), we can divide both sides by \( 3\sqrt{a} \) to find \( |x| = x \). This collapses the equation down to the simple requirement for \( x \) to be non-negative.
Knowing how and when to simplify expressions helps to not only solve equations effectively but also deeply understand the relationships between different algebraic terms.