Problem 110 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-x+3<0\) (b) \(2 x^{2}-x+3 \geq 0\)

Step-by-Step Solution

Answer

(a) No solution; (b) All real numbers, \(x \in \mathbb{R}\).

Step 1: Analyze the Quadratic Expression

Start by examining the quadratic expression in both inequalities: \(2x^2 - x + 3\). This expression does not factor nicely, so we'll consider its discriminant to understand the nature of its roots.

Step 2: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute \(a = 2\), \(b = -1\), and \(c = 3\) to find the discriminant: \(D = (-1)^2 - 4(2)(3) = 1 - 24 = -23\). Since \(D < 0\), the quadratic has no real roots and does not intersect the x-axis.

Step 3: Evaluate the Quadratic for Inequality (a)

For \(2x^2 - x + 3 < 0\), we need values of \(x\) that make the quadratic expression negative. Since the discriminant is negative and the coefficient of \(x^2\) is positive, the parabola opens upwards and stays above the x-axis for all real \(x\). Thus, \(2x^2 - x + 3 < 0\) has no solution.

Step 4: Evaluate the Quadratic for Inequality (b)

For \(2x^2 - x + 3 \geq 0\), we seek values of \(x\) for which the quadratic is non-negative. As previously mentioned, the parabola opens upwards and never dips below the x-axis. Therefore, \(2x^2 - x + 3\) is greater than or equal to zero for all real \(x\).

Key Concepts

Discriminant

The discriminant is a key component in understanding quadratic equations. It can be calculated using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients in the quadratic equation \(ax^2 + bx + c = 0\). The value of the discriminant gives us insight into the nature of the roots of the quadratic equation:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, often referred to as a repeated or double root.
• If \(D < 0\), there are no real roots, but instead, two complex roots.

In our example, the discriminant \(D = -23\), which means the quadratic equation \(2x^2 - x + 3 = 0\) has no real roots. This directly impacts how the quadratic behaves when considering inequalities, as it implies that the quadratic does not intersect the x-axis.

Parabola behavior

A quadratic equation in the form \(ax^2 + bx + c = 0\) represents a parabola on a graph. The behavior of this parabola is important when determining the solutions to inequalities involving the quadratic expression:

• If the coefficient \(a\) is positive, the parabola opens upwards. This means that it will have a minimum point somewhere on the graph.
• If the coefficient \(a\) is negative, the parabola opens downwards, having a maximum point instead.

In the exercise we are examining, the parabola described by the expression \(2x^2 - x + 3\) opens upwards because \(a = 2\) is positive. This indicates that the lowest point on the graph does not dip below the x-axis, particularly important when solving inequalities. This behavior helps explain why the quadratic remains non-negative over its entire range.

Quadratic equation solving

Solving quadratic equations focuses on finding the values of \(x\) when the quadratic expression equals zero, but inequalities extend this concept:

• For \(2x^2 - x + 3 < 0\), since the parabola opens upwards and the discriminant is negative (no real roots where it crosses the x-axis), the quadratic stays above the x-axis for all real \(x\). Thus, this inequality has no solutions.
• For \(2x^2 - x + 3 \geq 0\), because the quadratic does not dip below the x-axis (as indicated by its behavior and the value of the discriminant), it remains non-negative for all real \(x\). This means the inequality holds true across the entire set of real numbers.

Understanding these relationships is key to effectively solving quadratic inequalities without simply relying on graphing methods.