Finding Values Of K For Real Solutions In Quadratic Equations

In mathematics, quadratic equations play a crucial role, appearing in various fields and applications. A quadratic equation is a polynomial equation of the second degree, typically written in the standard form ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable. The solutions or roots of a quadratic equation are the values of x that satisfy the equation. Understanding the nature of these solutions—whether they are real or complex, distinct or repeated—is fundamental in algebra and beyond.

One key aspect of analyzing quadratic equations is determining the conditions under which the equation has real solutions. Real solutions are those that can be plotted on a number line, as opposed to complex solutions, which involve imaginary numbers. The existence and nature of real solutions are determined by the discriminant, a critical component derived from the coefficients of the quadratic equation. This article delves into the process of finding the values of k for which a given quadratic equation has two real solutions, focusing on the discriminant and its implications.

In the context of the quadratic equation 5x² + 3x + k = 0, our goal is to identify the range of values for the constant k that ensures the equation has two distinct real roots. This involves calculating the discriminant and setting up an inequality that reflects the condition for real solutions. By solving this inequality, we can pinpoint the specific values of k that meet the criteria. This exploration not only enhances our understanding of quadratic equations but also demonstrates the practical application of algebraic principles in problem-solving. Mastering these concepts is essential for anyone studying mathematics, as quadratic equations form the basis for more advanced topics and are frequently encountered in various mathematical and scientific contexts.

Understanding the Discriminant

To determine the conditions under which a quadratic equation has two real solutions, we must first understand the discriminant. The discriminant is a critical component of the quadratic formula, which is used to find the solutions (or roots) of a quadratic equation. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

In this formula, a, b, and c are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. The discriminant, often denoted as Δ (Delta), is the expression under the square root:

Δ = b² - 4ac

The discriminant plays a crucial role in determining the nature of the solutions of the quadratic equation. The value of the discriminant provides valuable information about whether the solutions are real, complex, distinct, or repeated. Specifically, there are three possible scenarios:

1. If Δ > 0, the quadratic equation has two distinct real solutions. This means there are two different values of x that satisfy the equation, and these values can be plotted on a number line. The square root of a positive number is a real number, leading to two different solutions due to the ± sign in the quadratic formula.
2. If Δ = 0, the quadratic equation has exactly one real solution (or a repeated real solution). In this case, the square root of zero is zero, and the quadratic formula simplifies to x = -b / (2a), yielding a single solution. This situation occurs when the parabola represented by the quadratic equation touches the x-axis at exactly one point.
3. If Δ < 0, the quadratic equation has two complex solutions. This means the solutions involve imaginary numbers because the square root of a negative number is an imaginary number. In this case, the solutions are complex conjugates, and they cannot be plotted on a standard number line.

For the quadratic equation to have two real solutions, the discriminant must be greater than zero. This condition ensures that the equation has two distinct real roots. Understanding and applying the discriminant is essential for solving quadratic equations and analyzing their solutions.

Applying the Discriminant to the Given Equation

Now, let's apply the concept of the discriminant to the given quadratic equation:

5x² + 3x + k = 0

In this equation, we can identify the coefficients as follows:

• a = 5
• b = 3
• c = k

To find the values of k for which the quadratic equation has two real solutions, we need to ensure that the discriminant (Δ) is greater than zero. As we established earlier, the discriminant is given by:

Δ = b² - 4ac

Substituting the coefficients from our equation, we get:

Δ = 3² - 4(5)(k) Δ = 9 - 20k

For the equation to have two real solutions, we need Δ > 0. Therefore, we set up the inequality:

9 - 20k > 0

Now, we solve this inequality for k. First, subtract 9 from both sides:

-20k > -9

Next, divide both sides by -20. Remember that when dividing by a negative number, we must reverse the inequality sign:

k < 9/20

This inequality tells us that the quadratic equation 5x² + 3x + k = 0 has two real solutions when k is less than 9/20. This is a critical result, as it provides a precise range of values for k that satisfy the condition for real solutions. In practical terms, any value of k less than 9/20 will result in the discriminant being positive, thus ensuring the quadratic equation has two distinct real roots. Understanding how to apply the discriminant in this manner is essential for solving a variety of problems involving quadratic equations and their solutions.

Solving the Inequality for k

In the previous section, we derived the inequality:

9 - 20k > 0

This inequality represents the condition for the quadratic equation 5x² + 3x + k = 0 to have two real solutions. To find the specific values of k that satisfy this condition, we need to solve the inequality. The process involves isolating k on one side of the inequality, following the standard rules of algebraic manipulation.

First, we subtract 9 from both sides of the inequality:

9 - 20k - 9 > 0 - 9 -20k > -9

This step simplifies the inequality by removing the constant term from the left side. Now, to isolate k, we need to divide both sides of the inequality by -20. It's crucial to remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign. Therefore, dividing both sides by -20 gives us:

(-20k) / (-20) < (-9) / (-20) k < 9/20

This is the solution to the inequality. It tells us that the values of k for which the quadratic equation has two real solutions are those that are less than 9/20. In decimal form, 9/20 is equal to 0.45, so the condition is that k must be less than 0.45. This means any value of k that is smaller than 0.45 will result in the discriminant being positive, thus ensuring the quadratic equation has two distinct real roots.

The solution k < 9/20 is not just a numerical result; it provides a clear and concise range of values for k. This range allows us to understand how the constant term k affects the nature of the solutions of the quadratic equation. By solving this inequality, we have successfully identified the condition for the equation to have two real solutions, which is a fundamental aspect of quadratic equation analysis.

Conclusion

In this exploration of the quadratic equation 5x² + 3x + k = 0, we set out to find the values of k for which the equation has two real solutions. By understanding and applying the concept of the discriminant, we successfully determined the condition that k must satisfy. The discriminant, given by the formula Δ = b² - 4ac, is a powerful tool for analyzing the nature of the solutions of a quadratic equation.

We began by reviewing the role of the discriminant in determining whether a quadratic equation has real, complex, distinct, or repeated solutions. We established that for an equation to have two real solutions, the discriminant must be greater than zero (Δ > 0). This condition ensures that the square root in the quadratic formula yields a real number, leading to two distinct roots.

Applying this principle to the given equation, we identified the coefficients a = 5, b = 3, and c = k. We then calculated the discriminant as:

Δ = 3² - 4(5)(k) = 9 - 20k

To find the values of k that result in two real solutions, we set up the inequality:

9 - 20k > 0

Solving this inequality, we found that:

k < 9/20

This result is the key takeaway of our analysis. It tells us that the quadratic equation 5x² + 3x + k = 0 has two real solutions for any value of k that is less than 9/20. This specific range of values ensures that the discriminant remains positive, which is the fundamental condition for real solutions.

In conclusion, understanding the discriminant and its application is essential for solving problems involving quadratic equations. By mastering these concepts, one can effectively analyze the nature of solutions and determine the conditions under which real solutions exist. The process outlined in this article provides a clear and concise method for finding the values of constants, such as k, that ensure a quadratic equation has two real solutions. This knowledge is invaluable in various mathematical and scientific contexts, making the study of quadratic equations a cornerstone of algebraic education.