Evaluating The Discriminant B² - 4ac For Quadratic Equations

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This article provides a comprehensive exploration of the expression b² - 4ac, commonly known as the discriminant, and its significance in determining the nature of roots in quadratic equations. We will evaluate this expression for various sets of values for a, b, and c, providing step-by-step calculations and insights into the results. The discriminant plays a crucial role in understanding the characteristics of quadratic equations, and by working through these examples, you will gain a solid understanding of its applications. Let's delve into the calculations and interpretations.

1. Understanding the Discriminant: b² - 4ac

The discriminant, represented by the expression b² - 4ac, is a fundamental component of the quadratic formula. The quadratic formula, given by x = (-b ± √(b² - 4ac)) / 2a, is used to find the roots (or solutions) of a quadratic equation in the form ax² + bx + c = 0. The discriminant, the part under the square root, holds the key to determining the nature of these roots. The value of the discriminant can tell us whether the quadratic equation has two distinct real roots, one real root (a repeated root), or two complex roots.

  • If b² - 4ac > 0: The quadratic equation has two distinct real roots. This means the parabola represented by the equation intersects the x-axis at two different points. These roots are real numbers, which can be rational or irrational.
  • If b² - 4ac = 0: The quadratic equation has one real root (a repeated or double root). In this case, the parabola touches the x-axis at exactly one point. The root is a real number.
  • If b² - 4ac < 0: The quadratic equation has two complex roots. This means the parabola does not intersect the x-axis. The roots are complex numbers, involving an imaginary part.

Understanding the discriminant allows us to quickly assess the type of solutions a quadratic equation will have without fully solving the equation. This is a powerful tool in various mathematical and scientific applications. In the following sections, we will apply this knowledge by evaluating the discriminant for several sets of values for a, b, and c, and interpreting the results.

2. Case 1: a = 1, b = 5, c = 4

In this first case, we are given a = 1, b = 5, and c = 4. Our task is to evaluate the discriminant, b² - 4ac, using these values. Substituting the given values into the expression, we have:

b² - 4ac = (5)² - 4(1)(4)

First, we calculate the square of b, which is 5²:

5² = 25

Next, we compute the product of 4, a, and c:

4(1)(4) = 16

Now, we substitute these results back into the discriminant expression:

b² - 4ac = 25 - 16

Finally, we subtract 16 from 25 to find the value of the discriminant:

25 - 16 = 9

So, the discriminant for this case is 9. Since 9 is a positive number (9 > 0), the quadratic equation with these coefficients will have two distinct real roots. This means the parabola represented by the equation will intersect the x-axis at two different points. These roots can be further calculated using the quadratic formula, but the discriminant alone tells us that they are real and distinct. This analysis demonstrates how the discriminant provides valuable information about the nature of the roots without the need for complete calculations.

3. Case 2: a = 2, b = 1, c = -1

For the second case, we are given the values a = 2, b = 1, and c = -1. We will now evaluate the discriminant, b² - 4ac, using these values. Substituting the given values into the expression, we get:

b² - 4ac = (1)² - 4(2)(-1)

First, we calculate the square of b, which is 1²:

1² = 1

Next, we compute the product of 4, a, and c:

4(2)(-1) = -8

Now, we substitute these results back into the discriminant expression:

b² - 4ac = 1 - (-8)

Subtracting a negative number is the same as adding its positive counterpart, so we have:

b² - 4ac = 1 + 8

Finally, we add 1 and 8 to find the value of the discriminant:

1 + 8 = 9

Thus, the discriminant for this case is 9. Similar to the first case, since 9 is a positive number (9 > 0), the quadratic equation with these coefficients will have two distinct real roots. This indicates that the parabola represented by the equation intersects the x-axis at two different points. Again, the discriminant provides a quick way to determine the nature of the roots, showing that they are real and distinct without needing to solve the full quadratic equation. This highlights the discriminant's utility in quickly analyzing quadratic equations.

4. Case 3: a = 4, b = 4, c = 1

In the third case, we have the values a = 4, b = 4, and c = 1. We will evaluate the discriminant b² - 4ac using these values. Substituting the given values into the expression, we obtain:

b² - 4ac = (4)² - 4(4)(1)

First, we calculate the square of b, which is 4²:

4² = 16

Next, we compute the product of 4, a, and c:

4(4)(1) = 16

Now, we substitute these results back into the discriminant expression:

b² - 4ac = 16 - 16

Finally, we subtract 16 from 16 to find the value of the discriminant:

16 - 16 = 0

Therefore, the discriminant for this case is 0. When the discriminant is equal to 0, the quadratic equation has exactly one real root (a repeated root). This means the parabola represented by the equation touches the x-axis at only one point. This root can be found by using the quadratic formula, but the discriminant already tells us that there is only one real solution. This case illustrates the discriminant's ability to identify quadratic equations with a single real solution, often referred to as a double root.

5. Case 4: a = 1, b = -2, c = -2

For the fourth case, we are given a = 1, b = -2, and c = -2. We will evaluate the discriminant, b² - 4ac, using these values. Substituting the given values into the expression, we get:

b² - 4ac = (-2)² - 4(1)(-2)

First, we calculate the square of b, which is (-2)²:

(-2)² = 4

Next, we compute the product of 4, a, and c:

4(1)(-2) = -8

Now, we substitute these results back into the discriminant expression:

b² - 4ac = 4 - (-8)

Subtracting a negative number is the same as adding its positive counterpart, so we have:

b² - 4ac = 4 + 8

Finally, we add 4 and 8 to find the value of the discriminant:

4 + 8 = 12

Thus, the discriminant for this case is 12. Since 12 is a positive number (12 > 0), the quadratic equation with these coefficients will have two distinct real roots. The parabola represented by the equation will intersect the x-axis at two different points. This demonstrates the discriminant's consistent ability to predict the nature of the roots, showing that they are real and distinct in this case.

6. Case 5: a = 9, b = 0, c = 1

In the fifth and final case, we have the values a = 9, b = 0, and c = 1. We will evaluate the discriminant b² - 4ac using these values. Substituting the given values into the expression, we obtain:

b² - 4ac = (0)² - 4(9)(1)

First, we calculate the square of b, which is 0²:

0² = 0

Next, we compute the product of 4, a, and c:

4(9)(1) = 36

Now, we substitute these results back into the discriminant expression:

b² - 4ac = 0 - 36

Finally, we subtract 36 from 0 to find the value of the discriminant:

0 - 36 = -36

Therefore, the discriminant for this case is -36. Since -36 is a negative number (-36 < 0), the quadratic equation with these coefficients will have two complex roots. This means the parabola represented by the equation does not intersect the x-axis. This case illustrates the discriminant's ability to identify when a quadratic equation has complex roots, which are not real numbers. The presence of complex roots indicates that the solutions involve imaginary numbers.

7. Conclusion: The Power of the Discriminant

In this article, we have thoroughly examined the discriminant (b² - 4ac) and its role in determining the nature of roots in quadratic equations. By evaluating the discriminant for five different sets of values for a, b, and c, we have demonstrated its ability to predict whether a quadratic equation has two distinct real roots, one real root (a repeated root), or two complex roots.

  • Case 1 (a = 1, b = 5, c = 4): Discriminant = 9, indicating two distinct real roots.
  • Case 2 (a = 2, b = 1, c = -1): Discriminant = 9, also indicating two distinct real roots.
  • Case 3 (a = 4, b = 4, c = 1): Discriminant = 0, indicating one real root (a repeated root).
  • Case 4 (a = 1, b = -2, c = -2): Discriminant = 12, indicating two distinct real roots.
  • Case 5 (a = 9, b = 0, c = 1): Discriminant = -36, indicating two complex roots.

These examples highlight the discriminant's efficiency and importance in quickly assessing the nature of solutions without fully solving the quadratic equation. Understanding the discriminant is crucial for anyone studying quadratic equations, as it provides valuable insights into the behavior and characteristics of these equations. The discriminant is a powerful tool in algebra and serves as a fundamental concept for further mathematical studies.