Problem 21 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(x^{2}+4 x+3=4\)

Step-by-Step Solution

Answer

The discriminant is 20, the equation has two distinct irrational roots, and the solutions are \(x = -2 \pm \sqrt{5}\).

Step 1: Rewrite the Equation

Start by rewriting the equation in the standard quadratic form: \(ax^2+bx+c=0\). Given equation is \(x^{2}+4x+3=4\). Subtract 4 from both sides to obtain: \(x^2 + 4x + 3 - 4 = 0\), which simplifies to \(x^2 + 4x - 1 = 0\). Here, \(a=1\), \(b=4\), and \(c=-1\).

Step 2: Find the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(D = 4^2 - 4 \cdot 1 \cdot (-1)\). Calculate: \(D = 16 + 4 = 20\).

Step 3: Analyze the Discriminant

Based on the value of the discriminant, we can conclude the nature of the roots. Since \(D = 20\) is positive and not a perfect square, there are two distinct irrational roots.

Step 4: Apply the Quadratic Formula

To find the exact solutions, use the quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(a=1\), \(b=4\), and \(D=20\) into the formula: \(x = \frac{-4 \pm \sqrt{20}}{2}\).

Step 5: Simplify the Solutions

Simplify \(\sqrt{20}\) to \(2\sqrt{5}\) because \(20 = 4 \times 5\) and \(\sqrt{4} = 2\). Therefore, \(x = \frac{-4 \pm 2\sqrt{5}}{2}\). Simplify the expression: \(x = -2 \pm \sqrt{5}\). This results in two solutions: \(x = -2 + \sqrt{5}\) and \(x = -2 - \sqrt{5}\).

Key Concepts

Discriminant

The discriminant is a key component in solving quadratic equations. It is represented by the letter \(D\) and is calculated using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation in standard form \(ax^2 + bx + c = 0\). The value of the discriminant gives important insights into the nature of the roots of the quadratic equation.

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, also known as a double root.
• If \(D < 0\), the equation has two complex roots which are not real.

In the solved exercise, the calculated discriminant is \(D = 20\). Since 20 is greater than zero and not a perfect square, it indicates that the quadratic equation has two distinct and irrational roots.

Quadratic Formula

The quadratic formula is a fundamental method for finding the roots of a quadratic equation when it cannot be easily factored. This formula is expressed as: \[x = \frac{-b \pm \sqrt{D}}{2a},\]where \(D\) is the discriminant. By substituting the values of \(a\), \(b\), and the previously computed \(D\) into the formula, we can find the exact solutions to the quadratic equation.
This method is versatile because it provides solutions even when roots are irrational or complex. In the exercise, using the quadratic formula with \(D = 20\) gives the roots as:

• \(x = \frac{-4 + \sqrt{20}}{2} \rightarrow x = -2 + \sqrt{5}\)
• \(x = \frac{-4 - \sqrt{20}}{2} \rightarrow x = -2 - \sqrt{5}\)

These roots confirm the distinct and irrational nature predicted by the discriminant.

Irrational Roots

Irrational roots are solutions to a quadratic equation that cannot be expressed as simple fractions. They are often expressed in terms of square roots when the discriminant is positive and not a perfect square. This means the roots involve terms like \(\sqrt{5}\) as seen in the example.
The characteristics of irrational roots include:

• They are always real numbers.
• They cannot be expressed as a simple ratio of two integers.
• For the equation given, the irrational roots are \(x = -2 + \sqrt{5}\) and \(x = -2 - \sqrt{5}\).

Understanding irrational roots is crucial as they often appear when solving real-world problems involving quadratic equations. It's important to know how to handle square roots to simplify and express these solutions accurately.