Problem 17 - 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. $$ x^{2}+3 x+1=0 $$

Step-by-Step Solution

Answer

The roots are irrational and unequal, and they are \( x = \frac{-3 \pm \sqrt{5}}{2} \).

Step 1: Recall the Discriminant Formula

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( D = b^2 - 4ac \). It helps determine the nature of the roots.

Step 2: Plug In the Coefficients

In the given equation \( x^2 + 3x + 1 = 0 \), the coefficients are \( a = 1 \), \( b = 3 \), and \( c = 1 \). Substitute these into the discriminant formula: \( D = 3^2 - 4(1)(1) \).

Step 3: Simplify to Find the Discriminant

Compute the value: \( D = 9 - 4 = 5 \). The discriminant \( D \) is 5.

Step 4: Determine the Nature of the Roots

Since the discriminant \( D = 5 \) is positive and not a perfect square, the roots are irrational and unequal.

Step 5: Find the Roots Using the Quadratic Formula

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be used to find the roots. Substitute \( a = 1 \), \( b = 3 \), and \( c = 1 \) into the formula: \( x = \frac{-3 \pm \sqrt{5}}{2} \).

Step 6: Simplify to Write the Roots

Simplified, the roots of the equation are \( x = \frac{-3 + \sqrt{5}}{2} \) and \( x = \frac{-3 - \sqrt{5}}{2} \).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that forms a parabola when graphed. These equations always have the highest degree of 2, which means they contain a term with the variable squared. In general form, a quadratic equation looks like this: \[ ax^2 + bx + c = 0 \] Here:

• \( a \), \( b \), and \( c \) are constants with \( a eq 0 \); \( a \) is the coefficient of the \( x^2 \) term.
• \( b \) is the coefficient of the \( x \) term.
• \( c \) is the constant term.

Quadratic equations appear in various real-world contexts, such as physics for describing projectile motion, in financial calculations, and much more. They can have a maximum of two solutions or roots.

Nature of Roots

The nature of the roots of a quadratic equation can be determined using its discriminant, denoted by \( D \). The discriminant is given by the formula: \[ D = b^2 - 4ac \] Here's how it helps in classifying the roots:

• If \( D > 0 \) and is a perfect square, the roots are rational and unequal.
• If \( D > 0 \) and is not a perfect square, the roots are irrational and unequal.
• If \( D = 0 \), the roots are rational and equal.
• If \( D < 0 \), the roots are not real numbers.

In the given exercise, \( D = 5 \), which is positive but not a perfect square, indicating that the roots are irrational and unequal. These observations are vital for solving and understanding quadratic equations.

Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. This formula derives from completing the square and is an essential tool in algebra. It is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here's how to use it: 1. Identify the coefficients \( a \), \( b \), and \( c \) from your quadratic equation. 2. Plug these values into the quadratic formula. 3. Calculate the value of the discriminant \( b^2 - 4ac \) to determine the nature of the roots, as previously discussed. 4. Solve for \( x \) using the positive and negative forms of the square root. In this example, substituting \( a = 1 \), \( b = 3 \), and \( c = 1 \) gives two roots: \[ x = \frac{-3 \pm \sqrt{5}}{2} \] This process easily finds the solutions even when the roots are complex or irrational, like in our case.

Irrational Roots

Irrational roots occur when the discriminant in a quadratic equation is a positive number that is not a perfect square. Such roots cannot be expressed as simple fractions, meaning they have non-repeating, non-terminating decimal forms. These roots usually involve the square root of a non-square number, leading to solutions in the form \( a \pm b\sqrt{n} \). In the current exercise, we concluded that the roots \( \frac{-3 + \sqrt{5}}{2} \) and \( \frac{-3 - \sqrt{5}}{2} \) are irrational. Irrational roots often appear in practical applications, especially in geometry and physics, where precise calculations are sometimes necessary. Understanding them helps in solving various real-world and theoretical problems efficiently.