Discriminant Analysis Of Quadratic Equation 3x² - 8x + 5 = 5x²

In the realm of mathematics, quadratic equations hold a fundamental position, particularly in algebra. These equations, characterized by their second-degree polynomial form, exhibit a rich array of properties and applications. One crucial aspect of understanding quadratic equations lies in determining the nature of their roots—whether they are real, distinct, or complex. This analysis is primarily conducted using the discriminant, a value derived from the coefficients of the quadratic equation. In this comprehensive article, we will delve into a specific quadratic equation, 3x² - 8x + 5 = 5x², to illustrate how the discriminant helps us ascertain the nature of its roots. We'll break down each step, ensuring clarity and depth in our exploration.

The Significance of the Discriminant

Before we dive into our specific equation, it's essential to grasp the role and calculation of the discriminant. For any quadratic equation in the standard form of ax² + bx + c = 0, the discriminant (${\Delta}$) is calculated using the formula:

${ \Delta = b² - 4ac }$

This simple yet powerful formula provides a wealth of information about the roots of the equation:

• If ${\Delta > 0}$, the equation has two distinct real roots.
• If ${\Delta = 0}$, the equation has exactly one real root (a repeated root).
• If ${\Delta < 0}$, the equation has no real roots; instead, it has two complex roots.

The discriminant, therefore, acts as a key indicator, helping us classify the solutions of a quadratic equation without actually solving for the roots themselves. This makes it an invaluable tool in mathematical analysis and problem-solving.

Transforming the Equation into Standard Form

The first step in analyzing the given equation, 3x² - 8x + 5 = 5x², is to transform it into the standard quadratic form ax² + bx + c = 0. This involves rearranging the terms to one side of the equation, which allows us to clearly identify the coefficients a, b, and c. Subtracting 5x² from both sides, we get:

${ 3x² - 5x² - 8x + 5 = 0 }$

Combining like terms, we simplify the equation to:

${ -2x² - 8x + 5 = 0 }$

Now, the equation is in the standard form, where we can easily identify a = -2, b = -8, and c = 5. This transformation is crucial because the discriminant formula relies on these coefficients.

Calculating the Discriminant

With the equation in standard form and the coefficients identified, we can proceed to calculate the discriminant using the formula:

${ \Delta = b² - 4ac }$

Substituting the values of a, b, and c, we get:

${ \Delta = (-8)² - 4(-2)(5) }$

${ \Delta = 64 + 40 }$

${ \Delta = 104 }$

The discriminant, ${\Delta}$, is 104. This value is positive, which immediately tells us something significant about the nature of the roots.

Interpreting the Discriminant Value

The value of the discriminant, ${\Delta = 104}$, is greater than 0. According to the principles of discriminant analysis, this indicates that the quadratic equation has two distinct real roots. This means there are two different real numbers that, when substituted for x in the equation, will satisfy the equation. The roots are real because the discriminant is positive, and they are distinct because the discriminant is not equal to zero.

Contrasting with Other Scenarios

To fully appreciate the significance of our result, let's briefly consider the other possible scenarios:

• If ${\Delta}$ had been 0, the equation would have had exactly one real root. This is the case of a repeated or double root, where the parabola represented by the quadratic equation touches the x-axis at exactly one point.
• If ${\Delta}$ had been less than 0, the equation would have had no real roots. Instead, it would have had two complex roots, which involve the imaginary unit i (where i² = -1). These roots do not appear on the real number line.

In our case, the positive discriminant clearly points to the existence of two real and distinct solutions, which provides a comprehensive understanding of the root characteristics for this specific equation.

Conclusion: The Nature of Roots in 3x² - 8x + 5 = 5x²

In summary, by transforming the equation 3x² - 8x + 5 = 5x² into the standard form -2x² - 8x + 5 = 0, we identified the coefficients necessary to calculate the discriminant. The calculated discriminant value of 104, being greater than 0, definitively tells us that the equation has two distinct real roots. This process underscores the importance of the discriminant in quadratic equation analysis, offering a straightforward method to determine the nature of roots without solving the equation explicitly.

Understanding the discriminant and its implications is a crucial skill in algebra, providing a foundation for more advanced mathematical concepts and applications. Whether in academic settings or practical problem-solving, the ability to quickly ascertain the nature of roots adds a layer of efficiency and insight to the mathematical process.

In conclusion, the statement about the equation 3x² - 8x + 5 = 5x² that is true is:

B. The discriminant is greater than 0, so there are two real roots.