Problem 66 - College Algebra

Question

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(4 x^{2}-2 x+3=0\)

Step-by-Step Solution

Answer

The discriminant of the quadratic equation is -44, which implies that the equation has no real solutions.

Step 1: Identify a, b and c

From the given equation \(4 x^{2}-2 x+3=0\), we can see that a = 4, b = -2 and c = 3.

Step 2: Compute the Discriminant

Substitute a = 4, b = -2 and c = 3 into the formula for the discriminant \(b^2 - 4ac\). Compute the value of the discriminant as follows: \((-2)^{2} - 4(4)(3) = 4 - 48 = -44\)

Step 3: Interpret The Discriminant

Because the value of the discriminant, -44, is less than 0, the given equation does not have real solutions.

Key Concepts

Discriminant

The discriminant is a key concept in quadratic equations, helping us to determine the nature of the solutions without actually solving the equation. The discriminant is derived from the quadratic formula and is expressed as \(b^2 - 4ac\). Here, \(a\), \(b\), and \(c\) are coefficients from the general form of the quadratic equation \(ax^2 + bx + c = 0\).

To find the discriminant:

• Identify the values of \(a\), \(b\), and \(c\) in the equation.
• Substitute these values into the formula \(b^2 - 4ac\).
• Compute the result to understand the nature of solutions the equation has.

The discriminant not only tells us whether solutions exist but also describes what kind they are. Its value directly affects whether solutions are real or complex.

Real and Complex Solutions

The solutions of a quadratic equation can be categorized into real or complex based on the discriminant. Understanding this helps predict the type of roots even before thoroughly solving the equation.

Types of Solutions

• Real and Distinct Solutions: If the discriminant is greater than 0, the equation has two distinct real solutions. This indicates two points where the parabola crosses the x-axis.
• Real and Equal Solutions: If the discriminant equals 0, there is exactly one real solution, meaning the parabola touches the x-axis at exactly one point (repeated root).
• Complex Solutions: If the discriminant is less than 0, no real solutions exist. Instead, the solutions are complex, appearing as conjugate pairs. This means the parabola does not intersect the x-axis at all.

Recognizing these possibilities helps you understand the graph behavior of a quadratic function even without plotting.

Quadratic Formula

The quadratic formula is invaluable for solving quadratic equations. It provides a means to find the solutions directly using the coefficients of the equation. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Here’s how it works:

• Use it to find solutions of the standard form equation \(ax^2 + bx + c = 0\).
• Identify the values of \(a\), \(b\), and \(c\) in the equation.
• Substitute these coefficients into the formula.
• Solve the resulting expressions for \(x\) to find the roots.

The expression under the square root, \(b^2 - 4ac\), is the discriminant, which we've discussed earlier. It informs us about the nature of the solutions. Depending on whether this discriminant is positive, zero, or negative, the roots can be real or complex. Understanding and applying this formula allows you to handle any quadratic equation efficiently.