Problem 22 - 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. $$ 2 x^{2}-3 x-5=0 $$

Step-by-Step Solution

Answer

The roots are rational and unequal; they are 2.5 and -1.

Step 1: Identify coefficients

For the quadratic equation \( ax^2 + bx + c = 0 \), identify \( a = 2 \), \( b = -3 \), and \( c = -5 \).

Step 2: Calculate the discriminant

The discriminant \( \Delta \) is given by the formula \( \Delta = b^2 - 4ac \). Substitute the values: \( \Delta = (-3)^2 - 4 \times 2 \times (-5) \).

Step 3: Simplify the discriminant expression

Calculate \( \Delta = 9 + 40 = 49 \).

Step 4: Interpret the discriminant

Since \( \Delta = 49 \) is a positive perfect square, the roots are rational and unequal.

Step 5: Use the quadratic formula

The roots of the quadratic equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \).

Step 6: Substitute the values in the formula

Substitute \( b = -3 \), \( \Delta = 49 \), and \( a = 2 \) into the formula: \( x = \frac{3 \pm \sqrt{49}}{4} \).

Step 7: Simplify the expression for roots

Calculate \( \sqrt{49} = 7 \), then \( x = \frac{3 \pm 7}{4} \).

Step 8: Calculate both possible roots

Solve for \( x \) to find the two roots: \( x_1 = \frac{3 + 7}{4} = 2.5 \) and \( x_2 = \frac{3 - 7}{4} = -1 \).

Key Concepts

Discriminant

In quadratic equations, the discriminant is a key tool to understand the nature of the roots. To find the discriminant, use the formula \( \Delta = b^2 - 4ac \) for the equation \( ax^2 + bx + c = 0 \). This value tells us whether the roots are:

• Real or not
• Rational or irrational
• Equal or unequal

A positive discriminant means the equation has two distinct real roots. If it's a perfect square, the roots will be rational and unequal. A zero discriminant indicates the roots are real and equal. Finally, a negative discriminant reveals the equation has no real roots. This is very helpful in predicting what type of solution we can expect before solving the equation.

Rational Roots

Rational roots are solutions of a quadratic equation that can be expressed as fractions of integers. This typically happens when the discriminant is a perfect square. In such cases, the square root can be exactly calculated, resulting in a neat, even-root.
To find these rational roots, you need to have:

• A positive discriminant that is a perfect square
• Coefficients that allow division resulting in whole numbers or neat fractions

For instance, in the problem above, the discriminant is 49, which is a perfect square. Thus, the roots of the given equation \( 2x^2 - 3x - 5 = 0 \) are rational, specifically \( x = 2.5 \) and \( x = -1 \), both fractions (\( \frac{5}{2} \) and \( \frac{-1}{1} \)). These neat solutions make understanding the nature of roots easier, especially in mathematical proofs or other applications.

Quadratic Formula

The quadratic formula is an invaluable tool in mathematics to solve quadratic equations. It is expressed as:\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]This formula provides the roots of any quadratic equation given the values of \( a \), \( b \), and \( c \). The term \( \pm \sqrt{\Delta} \) accounts for the two potential roots - one for the positive and one for the negative scenario.
To use the quadratic formula effectively:

• First calculate the discriminant \( \Delta \)
• Substitute \( a \), \( b \), and \( \Delta \) into the formula
• Simplify to find the roots

This formula is versatile as it works for all types of quadratic equations, whether the roots are real, rational, or irrational. Its universality makes it a cornerstone in algebra and calculus.

Real Roots

Real roots are solutions to quadratic equations that are real numbers, not imaginary. They can either be rational or irrational depending on the discriminant:
- **Rational Roots:** Occur when \( \Delta > 0 \) and is a perfect square- **Irrational Roots:** Occur when \( \Delta > 0 \) but is not a perfect square
If the discriminant equals zero, the equation has one real root that is repeated. This is known as a "double root." The formula \( x = \frac{-b}{2a} \) can still find this single root effectively, indicating the point at which the parabola vertex touches the x-axis. Conversely, if \( \Delta < 0 \), there are no real roots, signifying the parabola doesn't intersect the x-axis at all. Real roots play a crucial role in graphing quadratic functions and predicting their behavior in practical scenarios.