Problem 63 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises \(63-70\), use the discriminant to determine the number of real solutions of the equation. $$ x^{2}-6 x+5=0 $$

Step-by-Step Solution

Answer

In the given quadratic equation, \(x^2 - 6x + 5 = 0\), the coefficients are \(a = 1\), \(b = -6\), and \(c = 5\). The discriminant, \(Δ = b^2 - 4ac = (-6)^2 - 4(1)(5) = 36 - 20 = 16\), is positive. Therefore, there are two real solutions.

Step 1: Identify the coefficients a, b, and c

In the given quadratic equation, \(x^2 - 6x + 5 = 0\), we can see that the coefficients are \(a = 1\), \(b = -6\), and \(c = 5\).

Step 2: Calculate the discriminant Δ

Using the formula for the discriminant, \(Δ = b^2 - 4ac\), we can substitute the coefficients we have found in Step 1: \(Δ = (-6)^2 - 4(1)(5)\) \(Δ = 36 - 20\) \(Δ = 16\)

Step 3: Analyze the discriminant to determine the number of real solutions

Since the discriminant is positive (\(Δ = 16\)), this means that the quadratic equation has two real solutions.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental component of algebra, characterized by an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a \) is not equal to zero. The reason \( a \) cannot be zero is that this would eliminate the \( x^2 \) term, reducing the equation to a linear one. The quadratic equation represents a parabola when plotted on a graph, and the solutions to the equation are the points where the parabola crosses the x-axis.

To solve a quadratic equation, one can perform factoring, complete the square, or implement the quadratic formula, which is often the most direct method. It is essential for students to be comfortable working with these equations, as they appear frequently in various mathematical contexts, from simple problems to more complex applications in physics and engineering.

Real Solutions

When solving quadratic equations, we are often interested in finding the real solutions—these are the values of \( x \) that satisfy the equation. Real solutions are points where the graph of the quadratic equation intersects the x-axis. Based on the value of the discriminant (\( \Delta \)), which is calculated from the coefficients of the equation, we can predict the number of real solutions without actually solving the equation.

A positive discriminant indicates two distinct real solutions – our parabola crosses the x-axis at two points. A discriminant of zero points to one real solution, meaning the parabola touches the x-axis at a single point, known as a repeating root or a double root. Lastly, a negative discriminant suggests that there are no real solutions – the parabola does not intersect the x-axis at all and instead opens above or below it, depending on the coefficient \( a \)'s sign.

Quadratic Formula

The quadratic formula is the cornerstone for solving quadratic equations and is derived from completing the square of the general quadratic equation. It provides a clear-cut method to find the solutions for any quadratic equation. The formula is as follows: \[ x = \frac{{-b \pm \sqrt{\Delta}}}{{2a}} \], where \( \Delta \) denotes the discriminant \( b^2 - 4ac \).

The '±' sign in the formula indicates that two solutions are calculated: one with a '+', the other with a '-'. These solutions can be real or complex depending on the value of the discriminant. To apply the quadratic formula, one must identify the coefficients \( a \), \( b \), and \( c \) from the equation, substitute them into the formula, and then perform the necessary arithmetic operations to solve for \( x \). It serves as a powerful tool, ensuring that learners can always find the solutions to quadratic equations, even when factoring or graphing is not feasible.