Quadratic Equation: Discriminant Of 12

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In the realm of mathematics, understanding quadratic equations is fundamental to solving a vast array of problems. These equations, typically in the form of $ax^2 + bx + c = 0$, hold the key to modeling parabolas, projectile motion, and various optimization scenarios. A crucial aspect of analyzing a quadratic equation lies in its discriminant. The discriminant, denoted by $\Delta$ or $b^2 - 4ac$, provides invaluable information about the nature of the roots (solutions) of the equation. It tells us whether the roots are real and distinct, real and equal, or complex conjugates, without actually having to solve for them. This makes the discriminant a powerful shortcut in mathematical analysis. When a quadratic equation has a discriminant of 12, it signifies that the equation has two distinct real roots. This is because a positive discriminant indicates two unique real solutions. A discriminant of zero would mean exactly one real root (a repeated root), and a negative discriminant would imply two complex conjugate roots.

Let's delve into how we can identify a quadratic equation that yields a discriminant of 12. We are given four potential equations, and our task is to calculate the discriminant for each and determine which one matches the specified value. Remember, the formula for the discriminant is $\Delta = b^2 - 4ac$. We need to carefully identify the coefficients $a$, $b$, and $c$ in each equation and substitute them into the formula. It's important to pay close attention to the signs of the coefficients, as a small error in sign can lead to an incorrect result. The process involves straightforward arithmetic operations, but precision is key. By systematically applying the discriminant formula to each option, we can pinpoint the correct equation. This exercise not only tests our understanding of the discriminant but also reinforces our ability to perform accurate calculations with quadratic expressions. The quest to find the equation with a discriminant of 12 is a direct application of these core algebraic principles, highlighting the elegance and utility of mathematical tools in problem-solving. The nature of the roots is directly tied to the value of the discriminant: if $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, there is exactly one real root (a repeated root); and if $\Delta < 0$, there are two complex conjugate roots. Since we are looking for a discriminant of 12, which is positive, we expect two distinct real roots for the correct equation. This understanding is critical as we proceed to analyze each option provided. The ability to quickly assess the nature of roots using the discriminant is a valuable skill in algebra, saving time and effort compared to using the quadratic formula directly for root analysis. Therefore, mastering the calculation and interpretation of the discriminant is a key step in becoming proficient with quadratic equations.

Analyzing the Options

Let's meticulously examine each of the provided options to determine which quadratic equation has a discriminant of 12. We will apply the discriminant formula, $\Delta = b^2 - 4ac$, to each equation, carefully identifying the coefficients $a$, $b$, and $c$. Remember that the standard form of a quadratic equation is $ax^2 + bx + c = 0$. It's crucial to ensure that the equations are in this standard form before extracting the coefficients. If an equation is presented in a slightly different arrangement, we might need to rearrange it first to clearly identify $a$, $b$, and $c$. For instance, if an equation is written as $5x^2 - 3x + 1 = -2$, we would first move the constant term to the left side to get $5x^2 - 3x + 3 = 0$, making $a=5$, $b=-3$, and $c=3$. Precision in this step prevents common errors.

Option A: $0 = -x^2 + 8x + 2$ Here, $a = -1$, $b = 8$, and $c = 2$. Calculating the discriminant: $\Delta = b^2 - 4ac = (8)^2 - 4(-1)(2) = 64 - (-8) = 64 + 8 = 72$. Since $72 \neq 12$, Option A is not the correct equation.

Option B: $0 = 2x^2 + 6x + 3$ Here, $a = 2$, $b = 6$, and $c = 3$. Calculating the discriminant: $\Delta = b^2 - 4ac = (6)^2 - 4(2)(3) = 36 - 24 = 12$. This matches our target discriminant! Therefore, Option B is a potential candidate.

Option C: $0 = -x^2 + 4x + 1$ Here, $a = -1$, $b = 4$, and $c = 1$. Calculating the discriminant: $\Delta = b^2 - 4ac = (4)^2 - 4(-1)(1) = 16 - (-4) = 16 + 4 = 20$. Since $20 \neq 12$, Option C is not the correct equation.

Option D: $0 = 4x^2 + 2x + 1$ Here, $a = 4$, $b = 2$, and $c = 1$. Calculating the discriminant: $\Delta = b^2 - 4ac = (2)^2 - 4(4)(1) = 4 - 16 = -12$. Since $-12 \neq 12$, Option D is not the correct equation.

Conclusion

After systematically calculating the discriminant for each of the given quadratic equations, we found that Option B: $0 = 2x^2 + 6x + 3$ is the equation that yields a discriminant of 12. This demonstrates the practical application of the discriminant formula in identifying specific characteristics of quadratic equations. Understanding the discriminant is a cornerstone of algebra, providing insights into the nature and number of solutions a quadratic equation possesses. It's a tool that simplifies analysis and deepens our comprehension of mathematical relationships. The ability to calculate and interpret the discriminant is a valuable skill for anyone studying mathematics, from high school algebra to advanced calculus. It allows us to quickly categorize equations and predict the behavior of their solutions without the need for extensive computation. Whether you're solving problems in physics, engineering, or economics, the principles of quadratic equations and their discriminants will invariably appear. Mastering these concepts equips you with a powerful analytical toolkit. For further exploration into the fascinating world of quadratic equations and their properties, you can visit Khan Academy's Mathematics section or explore the detailed resources on Wolfram MathWorld's page on Quadratic Equations. These resources offer comprehensive explanations, examples, and practice problems to solidify your understanding. Keep practicing, and you'll soon find yourself navigating the complexities of quadratic equations with confidence and ease.