Problem 81 - College Algebra Essentials

Question

In Exercises \(75-82,\) compute the discriminant. Then determine the number and type of solutions for the given equation. $$ x^{2}-3 x-7=0 $$

Step-by-Step Solution

Answer

The discriminant is 37, so the equation has two distinct real solutions.

Step 1: Identify coefficients

Identify the coefficients a, b, and c in the given quadratic equation \(1x^2 - 3x - 7 = 0\). So, a=1, b=-3, and c=-7.

Step 2: Compute the Discriminant

Substitute the coefficients into the discriminant formula \(D = b^2 - 4ac\). This gives us \(D = (-3)^2 - 4*1*(-7) = 9 + 28 = 37.\

Step 3: Determine Number and Type of Solutions

Since the discriminant D > 0, the quadratic equation has two distinct real solutions.

Key Concepts

Discriminant

In the realm of quadratic equations, the discriminant plays a pivotal role. It helps us understand the nature of the solutions without actually solving the equation. The discriminant is denoted by the symbol \(D\), and it is calculated using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

This value tells us how many and what type of solutions the quadratic equation will have. It simplifies the process of analysis and gives us a clear picture just by looking at one number. Remember, quadratic equations can have different types of solutions depending on the value of the discriminant, making it an essential tool in algebra.

Number of Solutions

The number of solutions of a quadratic equation is directly influenced by the discriminant, \(D\). Depending on the value of \(D\), here is how it determines the number of solutions:

• If \(D > 0\), there are two distinct real solutions. This means the parabola associated with the quadratic equation intersects the x-axis at two separate points.
• If \(D = 0\), there is exactly one real solution or we can say the solutions are repeated. In this case, the parabola touches the x-axis at a single point, known as a tangent.
• If \(D < 0\), there are no real solutions, but instead there are two complex solutions. This scenario means the parabola does not intersect the x-axis at all.

This classification helps in predicting the behavior of the solutions without solving the equation entirely.

Real Solutions

Real solutions are the kind of solutions you normally encounter on the number line. In the context of quadratic equations, the discriminant helps us determine whether these solutions exist. When we talk about real solutions, we refer to the values of \(x\) that actualize in our physical number system without involving imaginary numbers.

For a given quadratic equation, if the discriminant \(D\) is greater than zero (\(D > 0\)), it indicates there are two distinct real solutions. These solutions can be found where the graph of the quadratic equation, which is a parabola, crosses the x-axis. This connection between algebraic solutions and graphical representations is why understanding the discriminant is so important in solving quadratic equations.

Coefficients in Quadratic Equations

Coefficients in quadratic equations are crucial components that determine the shape and position of the equation's graph on the coordinate plane. When you express a quadratic equation in the standard form \(ax^2 + bx + c = 0\), you identify three main parts:

• \(a\) is the coefficient of \(x^2\) and determines the "width" and "direction" of the parabola. If \(a\) is positive, the parabola opens upwards, and if negative, downwards.
• \(b\) is the coefficient of \(x\) and impacts the "tilt" of the parabola's axis.
• \(c\) is the constant term that affects the "vertical position" of the parabola on the graph.

Understanding these coefficients not only helps in calculating the discriminant but also in grasping the complete picture of how a quadratic equation behaves.