Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$ 3 x^{2}+5 x-8=0 $$

Step-by-Step Solution

Answer

The discriminant is 121; there are two distinct real solutions.

Step 1: Identify the Quadratic Equation Coefficients

The given quadratic equation is \(3x^2 + 5x - 8 = 0\). Identify the coefficients: \(a = 3\), \(b = 5\), and \(c = -8\).

Step 2: Calculate the Discriminant

The formula for the discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is \(D = b^2 - 4ac\). Substitute the coefficients: \(D = 5^2 - 4 \times 3 \times (-8)\).

Step 3: Simplify the Discriminant Expression

Calculate \(5^2 = 25\) and \(-4 \times 3 \times (-8) = 96\). Therefore, \(D = 25 + 96\).

Step 4: Evaluate the Discriminant

Add the values: \(D = 25 + 96 = 121\). The discriminant is 121.

Step 5: Determine the Nature of the Solutions

Since the discriminant \(D = 121\) is greater than zero and a perfect square, the quadratic equation has two distinct real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form \(ax^2 + bx + c = 0\). In this equation:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

Quadratic equations are very common in algebra, and they graph a parabola when plotted on a coordinate plane. Understanding quadratics is essential because they appear in many areas of mathematics and real-world applications. Solving quadratic equations often involves finding values of \(x\) (known as the roots or solutions) that make the equation true.

Coefficients

Coefficients in a quadratic equation play a crucial role in determining its properties and solutions. The coefficients are the numerical values placed before the variables in the equation \(ax^2 + bx + c = 0\). Each coefficient impacts the parabola represented by the quadratic in distinct ways:

• **\(a\) (leading coefficient):** Determines the direction and width of the parabola's opening.
• **\(b\):** Affects the symmetry and position of the parabola on the graph.
• **\(c\):** Represents the point where the parabola intersects the y-axis.

For example, in the equation \(3x^2 + 5x - 8 = 0\), \(a = 3\), \(b = 5\), and \(c = -8\). These coefficients let us apply the quadratic formula and find the solutions.

Nature of Solutions

The nature of solutions to a quadratic equation is determined by calculating the discriminant, denoted as \(D\). The discriminant is calculated using the formula: \(D = b^2 - 4ac\). By analyzing the value of \(D\), we can predict the number and type of solutions without actually solving the equation:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution (a repeated or double root).
• If \(D < 0\), there are no real solutions, but two complex solutions.

In our example, \(D = 121\), which is greater than zero and indicates two distinct real solutions. Knowing these conditions helps students understand the possible outcomes just by evaluating \(D\), simplifying the problem-solving process.

Perfect Square

A perfect square is a number that can be expressed as the square of an integer. In the context of quadratic equations, when the discriminant \(D\) is a perfect square, it typically signifies that the quadratic equation's solutions are rational numbers. For instance, if \(D = 121\), which equals \(11^2\), this means 121 is a perfect square. This ensures that any solutions derived will be in a simpler, more comprehendible form, typically rational. Identifying whether \(D\) is a perfect square is valuable because it gives additional information about the precision of the solutions. In our example, knowing \(D = 121\) indicates two distinct rational solutions, making it easier to predict the nature of the equation's roots.