Problem 49 - Beginning and Intermediate Algebra

Question

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ x^{2}+4 x+2=0 $$

Step-by-Step Solution

Answer

The solutions are C. two irrational numbers. Use the quadratic formula.

Step 1: Identify the coefficients

In the quadratic equation \(x^{2} + 4x + 2 = 0\), the coefficients are: \(a = 1\), \(b = 4\), and \(c = 2\).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\Delta = b^2 - 4ac\). Substituting the values of \(a\), \(b\), and \(c\) into the formula: \(\Delta = 4^2 - 4 \cdot 1 \cdot 2 = 16 - 8 = 8\).

Step 3: Determine the nature of the solutions

Since the discriminant \(\Delta = 8\) is greater than 0 and not a perfect square, the equation has two irrational solutions. Therefore, the solutions are C. two irrational numbers.

Step 4: Method to solve the equation

Since the quadratic equation does not factorize easily, the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\) should be used to solve the equation.

Key Concepts

Quadratic Formula

A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\). The quadratic formula is a powerful tool to solve these equations: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. To use the formula, we need the values of \(a\), \(b\), and \(c\).
These values are the coefficients of the quadratic equation. This formula can solve any quadratic equation, whether it has real or complex solutions.
Steps to use the quadratic formula:

• Identify coefficients \(a\), \(b\), and \(c\).
• Plug them into the formula.
• Calculate the discriminant (\(\Delta = b^2 - 4ac\)) within the formula.
• Simplify the expression to find the values of \(x\).

Rational and Irrational Solutions

The solutions of a quadratic equation depend on its discriminant (\(\Delta\)). The discriminant tells us the nature of the roots:

• If \(\Delta = 0\), the equation has one real and rational solution.
• If \(\Delta > 0\) and is a perfect square, there are two real and rational solutions.
• If \(\Delta > 0\) but not a perfect square, the solutions are real but irrational.
• If \(\Delta < 0\), the solutions are complex (non-real).

For example, in the equation \((x^2 + 4x + 2 = 0)\), since \(\Delta = 8\), which is greater than 0 and not a perfect square, the equation has two irrational solutions.

Zero-Factor Property

The zero-factor property is a principle used to solve quadratic equations by factoring. It states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both).
This property applies when you can express the equation in the form of two binomials multiplied together.
For instance, \((x^2 + 4x + 2)\) doesn't factorize easily, so the zero-factor property is not ideal here. Instead, the quadratic formula is preferable.

Discriminant

The discriminant (\(\Delta\)) is a key part of the quadratic formula and determines the nature of the roots. It’s calculated as: \[ \Delta = b^2 - 4ac \].
Let’s break down how to find this:

• Identify \(a\), \(b\), and \(c\) from the quadratic equation.
• Substitute these values into the formula to compute \(\Delta\).

For example, in the equation \((x^2 + 4x + 2 = 0)\), the coefficients are \(a = 1\), \(b = 4\), and \(c = 2\). Substituting these into the formula gives \[ \Delta = 4^2 - 4 \cdot 1 \cdot 2 = 16 - 8 = 8 \].
Since \(\Delta = 8\) is greater than 0 and not a perfect square, the solutions are two irrational numbers.