Problem 69 - A Graphical Approach to College Algebra

Question

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$x^{2}+8 x+16=0$$

Step-by-Step Solution

Answer

The equation has one real rational solution.

Step 1: Identify the Quadratic Coefficients

In the quadratic equation form \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 1\), \(b = 8\), and \(c = 16\).

Step 2: Calculate the Discriminant

The discriminant \(D\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula to get \(D = 8^2 - 4 \times 1 \times 16 = 64 - 64 = 0\).

Step 3: Interpret the Discriminant

A discriminant of \(D = 0\) indicates that the quadratic equation has exactly one real solution. This solution is also rational since it results from a perfect square.

Key Concepts

Understanding Real Solutions

When dealing with quadratic equations, an important question is how many real solutions the equation has. A real solution refers to a value of the variable that satisfies the equation without involving any imaginary numbers. In the context of a quadratic equation, a real solution means that the parabola representing the equation intersects the x-axis at one or more points. You can determine the nature of the solutions of a quadratic equation using the discriminant, which is calculated from the coefficients of the equation. Depending on the value of the discriminant:

• If it's positive, there are two distinct real solutions.
• If it's zero, there is exactly one real solution.
• If it's negative, there are no real solutions, only complex ones.

In our given equation, since the discriminant is zero, it confirms that there is one real solution.

Basics of a Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It generally takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(ae0\). The solutions to this type of equation can be found using various methods such as factoring, completing the square, or using the quadratic formula. The quadratic formula allows us to find the roots of the equation by calculating:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The portion under the square root, \(b^2 - 4ac\), is called the discriminant. This part determines the number and type of solutions you might expect.
Quadratic equations are frequently solved in various mathematical problems because they model many natural phenomena, from physics to finance.

What Makes Solutions Rational or Irrational

When discussing the solutions of an equation, determining whether they are rational or irrational is crucial. A rational solution is one that can be written as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(qeq0\). In contrast, irrational solutions cannot be expressed as such a simple fraction, often involving square roots that do not simplify to a whole number.For our quadratic equation, the discriminant plays a significant role in determining if the solution is rational or irrational:

• If the discriminant \(b^2 - 4ac\) is a perfect square, the solutions of the equation are rational.
• Otherwise, the solutions are irrational.

Since our equation's discriminant is zero, it implies a single real and rational solution. This solution emerges from the double root created when the parabola just touches the x-axis.