Problem 76 - Algebra and Trigonometry

Question

Determine the values of the variable for which the expression is defined as a real number. $$ \sqrt{3 x^{2}-5 x+2} $$

Step-by-Step Solution

Answer

The expression is defined for \( x \le \frac{2}{3} \) or \( x \ge 1 \).

Step 1: Identify the Expression Under the Square Root

The given expression is \( \sqrt{3x^2 - 5x + 2} \). For the expression to be defined as a real number, the value inside the square root, \( 3x^2 - 5x + 2 \), must be non-negative.

Step 2: Set Up the Inequality

To ensure the expression under the square root is non-negative, we need to solve the inequality: \( 3x^2 - 5x + 2 \ge 0 \).

Step 3: Solve for the Roots of the Quadratic Equation

First, find the roots of the quadratic equation \( 3x^2 - 5x + 2 = 0 \). Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3, b = -5, c = 2 \), we calculate:\[x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{5 \pm \sqrt{25 - 24}}{6} = \frac{5 \pm 1}{6}\]This gives the roots \( x = 1 \) and \( x = \frac{2}{3} \).

Step 4: Determine Intervals for the Quadratic Inequality

Using the roots \( x = \frac{2}{3} \) and \( x = 1 \), evaluate the intervals determined by these points: \(( -\infty, \frac{2}{3} ), (\frac{2}{3}, 1), (1, \infty)\). Check the sign of the quadratic expression \( 3x^2 - 5x + 2 \) within each interval.

Step 5: Test the Intervals

Plug test points from each interval into the inequality:- For \(( -\infty, \frac{2}{3} )\), test \( x = 0 \): \( 3(0)^2 - 5(0) + 2 = 2 \), positive.- For \((\frac{2}{3}, 1)\), test \( x = 0.8 \): \( 3(0.8)^2 - 5(0.8) + 2 = 0.32 - 4 + 2 = -0.48 \), negative.- For \((1, \infty)\), test \( x = 2 \): \( 3(2)^2 - 5(2) + 2 = 12 - 10 + 2 = 4 \), positive.

Step 6: Conclusion on Defined Intervals

Based on the sign tests, the quadratic expression is non-negative on the intervals \(( -\infty, \frac{2}{3} ] \) and \([1, \infty)\). Therefore, the expression \( \sqrt{3x^2 - 5x + 2} \) is defined as a real number on these intervals.

Key Concepts

Real Numbers

Real numbers are values that encompass both rational and irrational numbers. They form the basis of algebra and can be thought of as any point along the number line. When we deal with square root expressions, the concept of real numbers is essential because the square roots of negative numbers are not real. They are classified as imaginary numbers when they can't be defined in the real number system.
Understanding real numbers helps us dictate where expressions like \( \sqrt{3x^2 - 5x + 2} \) are defined. For this expression to be a real number, the value inside the square root must be greater than or equal to zero. This requirement stems from the property that square roots of non-negative numbers are in the realm of real numbers.
In summary, identifying when an expression is a real number hinges on checking if values fall within the domain of real numbers by ensuring non-negativity in contexts involving square roots.

Square Root Function

The square root function, denoted as \( \sqrt{x} \), involves finding a number that, when multiplied by itself, provides the original number \( x \). It is one of the most fundamental functions in mathematics, serving as a building block for other complex operations.
One must hold a particular condition: the number \( x \) within a square root must be zero or positive for the expression to result in real numbers. For example, in the inequality \( 3x^2 - 5x + 2 \ge 0 \), we ensure that \( 3x^2 - 5x + 2 \) remains non-negative for real solutions. This means solving a quadratic inequality that requires us to determine intervals where the inequality holds true.
When working with a square root function, pay attention to the expressions under the square root to distinguish where they define real values. This approach is vital in problems involving quadratic expressions and their respective inequalities.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula provides solutions for the variable \( x \) by computing directly from the coefficients \( a \), \( b \), and \( c \).
The quadratic formula relies heavily on the discriminant \( b^2 - 4ac \). This portion of the formula reveals crucial information about the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (repeated).
• If the discriminant is negative, there are no real roots, indicating the solutions are complex or imaginary.

Applying the quadratic formula in our example allowed us to quickly find the values at which the expression becomes zero, specifically \( x = 1 \) and \( x = \frac{2}{3} \). These roots helped in evaluating intervals on which the original quadratic inequality (and thus the square root) holds as non-negative. Understanding the quadratic formula and its implications is crucial for solving quadratic inequalities efficiently.