Problem 36 - Algebra 2

Question

Evaluate the discriminant of each equation. Tell how many solutions each equation has and whether the solutions are real or imaginary. $$ 6 x^{2}-2 x+5=0 $$

Step-by-Step Solution

Answer

The discriminant is \(D = -116\). The equation has two imaginary solutions.

Step 1: Identifying the coefficients

Recognize the standard form of a quadratic equation, which is given by \(ax^2 + bx + c = 0\text{\). In the equation \(6x^2 - 2x + 5 = 0\), the coefficients are identified as \(a = 6\), \(b = -2\), and \(c = 5\).

Step 2: Calculating the discriminant

Calculate the discriminant \(D\) using the formula \(D = b^2 - 4ac\). Substitute the identified coefficients into the formula to get \(D = (-2)^2 - 4(6)(5) = 4 - 120 = -116\).

Step 3: Determining the number and type of solutions

Assess the discriminant's value to determine the number and nature of solutions. Since the discriminant is negative \(D = -116 < 0\), there are two imaginary solutions.

Key Concepts

Quadratic Equations

Quadratic equations are an essential concept in algebra that students encounter frequently. These equations take the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the unknown variable we are trying to solve for. The highest exponent in the equation is 2, which is what makes it 'quadratic' (from the Latin 'quadratum' for square).

Understanding how to solve quadratic equations is vital because they appear in various practical situations, from calculating areas to predicting projectile paths. Solving a quadratic equation means finding the values of \(x\) that make the equation true, which could lead to one, two, or no real solutions, or two imaginary solutions.

Discriminant Analysis

Discriminant analysis in the context of quadratic equations relates to the discriminant's role in determining the nature and number of solutions to the equation. The discriminant, often represented by the symbol \(D\), is calculated using the coefficients of the quadratic equation with the formula \(D = b^2 - 4ac\).

The outcome of the discriminant provides critical information: if \(D > 0\), there are two distinct real solutions; if \(D = 0\), there is exactly one real solution (also known as a repeated or double root); and if \(D < 0\), the equation has two complex solutions, commonly referred to as imaginary solutions because they involve the square root of a negative number. Hence, the discriminant is a powerful tool for preemptively assessing the solutions without solving the entire equation.

Real and Imaginary Solutions

Quadratic equations can have either real or imaginary solutions, and the discriminant is the key to knowing which type you'll encounter. Real solutions can be visualized as points where a parabola (the graph of a quadratic equation) intersects the x-axis. They are the x-values you can typically plot on a standard graph.

On the other hand, imaginary solutions arise when the parabola does not intersect the x-axis at all. Instead of finding points on the graph, the solutions include the square root of a negative number, which cannot be plotted on a real number line. Real solutions are typically preferred in real-world applications because imaginary numbers, although valuable in advanced mathematics and physics, do not have a direct visual representation.

Quadratic Formula

The quadratic formula is a reliable method for finding the solutions to a quadratic equation, regardless of whether they are real or imaginary. It is expressed as \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). This formula utilizes the coefficients from the quadratic equation \(ax^2 + bx + c = 0\) to find the values of \(x\).

The symbol \(\pm\) indicates that there will be two solutions, one involving addition and the other subtraction. The quadratic formula provides a precise way of calculating these solutions in a finite number of steps—hence it is a fundamental tool in algebra. In using this formula, the discriminant \(\sqrt{{b^2 - 4ac}}\) within the formula becomes a clear indicator of the nature of the solutions.