Problem 31 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(-\frac{1}{2} x^{2}+x+3=0\)

Step-by-Step Solution

Answer

The given equation has two distinct real solutions.

Step 1: Identify the values of a, b, and c.

The given equation is \(-\frac{1}{2} x^{2}+x+3=0\) and it is in the form of a quadratic equation \(ax^{2} + bx + c = 0\). From this, we can identify \(a = -\frac{1}{2}\), \(b = 1\), and \(c = 3\).

Step 2: Calculate the Discriminant

The discriminant is calculated as \(b^{2} - 4ac\). Substituting the values from step 1, we have \(D = 1^{2} - 4 (-\frac{1}{2} * 3) = 1 + 6 = 7\).

Step 3: Determine the number of solutions

Now, compare the discriminant value with zero. As the discriminant \(D = 7 > 0\), the given quadratic equation has two distinct real solutions.

Key Concepts

Discriminant

In the realm of quadratic equations, the discriminant is a key player in determining the nature of the solutions. It is represented by the symbol 'D' in equations. The discriminant is obtained from the coefficients of the quadratic equation, which is standardly expressed as \( ax^2 + bx + c = 0 \), where \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \) and \( c \) is the constant term.

The formula to calculate the discriminant is \( D = b^2 - 4ac \). This value tells us not just the quantity, but the nature of the solutions:

• 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.
• If \( D < 0 \), there are no real solutions; instead, there are two complex solutions.

In our example, \( D = 7 \) which indicates there are two real and distinct solutions to the equation \( -\frac{1}{2} x^{2}+x+3=0 \).

Real Solutions of Quadratic Equations

A quadratic equation may yield different types of real solutions depending on its discriminant. As we've noted, the sign of the discriminant determines the number and nature of the solutions. When a quadratic equation has a positive discriminant, this means the parabola (the graph of the equation) intersects the x-axis at two points, yielding two real solutions.

These solutions are the x-coordinates of the points where the graph intersects the x-axis and can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{D}}{2a} \). When \( D > 0 \), the '±' symbol in the quadratic formula indicates that there are two different solutions obtained by 'adding' and 'subtracting' the square root of the discriminant. For the equation in our example, since \( D = 7 \), we can confirm that we will find two unique solutions by applying the quadratic formula.

Solving Quadratic Equations

To solve quadratic equations like \( -\frac{1}{2} x^{2}+x+3=0 \) several methods are available depending on the given information and equation structure. The most common methods include:

• Factoring the quadratic
• Completing the square
• Using the quadratic formula

Each method has its particular use cases. Factoring comes in handy when the equation can be easily decomposed into two binomials. Completing the square is a method that allows equations to be written in a perfect square trinomial, leading to a solution that involves taking square roots. The quadratic formula, however, is the most universal method, as it can be used for all quadratic equations and is derived directly from the process of completing the square.

The quadratic formula is \( x = \frac{-b \pm \sqrt{D}}{2a} \) and, as we've already established, it provides solutions based upon the value of the discriminant. Applying it to our original problem, we would substitute in the coefficients \( a \) as \( -\frac{1}{2} \), \( b \) as \(1\), and \( c \) as \(3\) to find the real solutions of the equation.