Problem 15 - Algebra 2 and Trigonometry

Question

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ x^{2}-12 x+36=0 $$

Step-by-Step Solution

Answer

The discriminant is 0, indicating rational and equal roots; both roots are 6.

Step 1: Understanding the Problem

We need to find the value of the discriminant of the quadratic equation \(x^2 - 12x + 36 = 0\) and determine the nature of the roots based on the discriminant.

Step 2: Identifying the Coefficients

For the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 1\), \(b = -12\), \(c = 36\).

Step 3: Calculating the Discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute the values: \(D = (-12)^2 - 4(1)(36)\).

Step 4: Solving for the Discriminant Value

Calculate the expression: \((-12)^2 = 144\) and \(4 \times 1 \times 36 = 144\). Therefore, \(D = 144 - 144 = 0\).

Step 5: Determining the Nature of the Roots

The value of the discriminant \(D = 0\), which means the quadratic equation has rational and equal roots.

Step 6: Finding the Roots

Since the discriminant is zero, we can find the roots using the formula for the roots \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(b = -12\), \(a = 1\), and \(\sqrt{D} = 0\): \(x = \frac{12}{2} = 6\). Both roots are equal to 6.

Key Concepts

Discriminant

In quadratic equations, one important concept to grasp is the discriminant. The discriminant is found using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the terms \(ax^2 + bx + c = 0\). It is a key component in determining the nature of the roots of a quadratic equation.

By calculating the discriminant, we can predict whether the roots are real or complex, and whether they are distinct or the same. This simple computation provides a wealth of information about the equation without having to solve it completely. Consider it as a diagnostic tool.

For example, in the quadratic equation \(x^2 - 12x + 36 = 0\), the coefficients \(a = 1\), \(b = -12\), and \(c = 36\) can be easily plugged into the formula, resulting in a discriminant of zero.

Nature of Roots

The nature of the roots in a quadratic equation is determined by the value of the discriminant. The discriminant not only helps identify whether the roots are real or not, but also provides information on their uniqueness and nature (whether rational or irrational).

Here is a quick guide to interpreting the discriminant:

• When \(D > 0\), the equation has two distinct real roots. If \(D\) is a perfect square, these roots are rational.
• When \(D = 0\), the equation has exactly one real root, which appears twice. These roots are rational and equal.
• When \(D < 0\), the roots are not real—they are complex or imaginary.

For our given equation with a discriminant of zero, it indicates the roots are real, rational, and identical.

Real Roots

Real roots are an outcome of positive or zero discriminants. If the discriminant yields a positive value or zero, the roots of the quadratic equation will be real numbers. Real roots can be readily identified and plotted on the number line. They represent the intersection of the parabola of the equation with the x-axis.

In contrast, a negative discriminant means the roots are not real; instead, the equation has complex roots which are not visible on the real number line. For the equation \(x^2 - 12x + 36 = 0\), the discriminant is 0, indicating the roots are real and indeed observable on the x-axis.

Rational Roots

Rational roots occur when the discriminant is a perfect square, including zero, and the roots can then be expressed as a simple fraction of integers. In the quadratic equation \(x^2 - 12x + 36 = 0\), we found that the discriminant is zero.

According to the nature of the roots as determined by the discriminant, this implies that the equation has

• Rational roots, as they are derived from basic arithmetic operations on integers
• The equation has two equivalent roots, making them not only rational but also equal.

These characteristics make such equations particularly easy to solve since both roots are simply the same number, which in this case is 6.