Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Use the discriminant to determine the number of real solutions of the quadratic equation. \(2 x^{2}-5 x=-5\)

Step-by-Step Solution

Answer

The quadratic equation has no real solutions as the discriminant is negative (\(-15\))

Step 1: Rearrange the equation

Start by rewriting the equation into standard quadratic form \(ax^2 + bx + c = 0\). Our given equation is \(2x^2 - 5x = -5\). Adding 5 to both sides gives \(2x^2 - 5x + 5 = 0\). Now from this equation, we can extract the coefficients \(a\), \(b\), and \(c\).

Step 2: Identify coefficients

From the quadratic equation, we clearly see that \(a = 2\), \(b = -5\), and \(c = 5\).

Step 3: Calculate discriminant

The discriminant \(D\) is given by the formula \(D = b^2 - 4 a c\). Substituting the identified coefficients, we find that \(D = (-5)^2 - 4 * 2 * 5 = 25 - 40 = -15\).

Step 4: Interpret the discriminant

The discriminant, \(D\), is negative, namely \(-15\). This means that the quadratic equation has no real solutions since a negative value under the square root (implied by the quadratic formula) results in an imaginary number.

Key Concepts

Discriminant

In solving quadratic equations, the discriminant plays a crucial role. It helps us determine the nature of the roots, or solutions, of the equation. For a quadratic equation in standard form, i.e., \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula:

• \(D = b^2 - 4ac\)

The value of the discriminant indicates the number and type of solutions. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, there is exactly one real solution. On the other hand, if it's negative, like in the given example where \(D = -15\), there are no real solutions and instead, the solutions are complex numbers. Understanding the discriminant is key to predicting the behavior of quadratic equations without actually solving them.

Real Solutions

When tackling quadratic equations, especially those in standard form, we're often interested in finding real solutions. These are the values of \(x\) that satisfy the equation, and they occur where the graph of the equation crosses the \(x\)-axis. Real solutions can number from zero to two:

• If the discriminant \(D > 0\), there are two distinct real solutions, meaning two \(x\)-intercepts on the graph.
• If \(D = 0\), there is exactly one real solution, indicating the graph tangentially touches the \(x\)-axis at one point.
• If \(D < 0\), as in our example, there are no real solutions because the graph does not intersect the \(x\)-axis.

Understanding this helps in visualizing how the graph of a quadratic equation behaves relative to the \(x\)-axis based just on the discriminant.

Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This form is necessary for calculating the discriminant and solving the equation using the quadratic formula. Converting an equation to standard form involves ensuring all terms are on one side of the equation, leaving zero on the other side.
For instance, in the given exercise, the equation \(2x^2 - 5x = -5\) was converted to \(2x^2 - 5x + 5 = 0\) to identify the coefficients \(a = 2\), \(b = -5\), and \(c = 5\). This setup facilitates analyzing the equation's characteristics and finding its solutions using the discriminant or other methods. Standard form is the foundational step to solving quadratic equations efficiently.