Determine Real Solutions Using The Discriminant For 16x^2 + 4 = 16x

In mathematics, especially when dealing with quadratic equations, determining the nature and number of solutions is a fundamental task. A powerful tool that aids in this determination is the discriminant. This article will delve into how the discriminant is used to find the number of real solutions for a given quadratic equation. We'll use the example equation $16x^2 + 4 = 16x$ to illustrate the process.

What is the Discriminant?

The discriminant is a part of the quadratic formula that reveals the nature of the roots (solutions) of a quadratic equation. A quadratic equation is generally expressed in the form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and x is the variable. The quadratic formula, used to find the solutions of the equation, is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant is the expression under the square root in the quadratic formula:

$D = b^2 - 4ac$

The value of the discriminant, D, tells us about the nature and number of solutions:

• If $D > 0$, the equation has two distinct real solutions.
• If $D = 0$, the equation has exactly one real solution (a repeated root).
• If $D < 0$, the equation has no real solutions; instead, it has two complex solutions.

Understanding the discriminant allows us to quickly determine the type of solutions without fully solving the quadratic equation.

Applying the Discriminant to $16x^2 + 4 = 16x$

Now, let's apply this knowledge to the equation $16x^2 + 4 = 16x$. The first step is to rewrite the equation in the standard quadratic form, $ax^2 + bx + c = 0$. To do this, subtract $16x$ from both sides:

$16x^2 - 16x + 4 = 0$

Now we can identify the coefficients:

• $a = 16$
• $b = -16$
• $c = 4$

Next, we calculate the discriminant using the formula $D = b^2 - 4ac$:

$D = (-16)^2 - 4(16)(4)$ $D = 256 - 256$ $D = 0$

Since the discriminant $D = 0$, the quadratic equation has exactly one real solution. This means the parabola represented by the equation touches the x-axis at only one point.

In summary, by calculating the discriminant, we can efficiently determine the number and type of solutions a quadratic equation possesses. This method saves time and provides valuable insights into the nature of the roots without needing to solve the entire equation.

Steps to Determine the Number of Real Solutions Using the Discriminant

To effectively use the discriminant, following a structured approach is essential. This ensures accuracy and clarity in determining the nature and number of real solutions for any quadratic equation. Here’s a step-by-step guide:

1. Rewrite the Quadratic Equation in Standard Form: The first step is to express the given quadratic equation in the standard form, which is $ax^2 + bx + c = 0$. This form is crucial because the coefficients a, b, and c are used in the discriminant formula. Rearranging the equation might involve moving terms from one side to the other, combining like terms, or setting the equation equal to zero.

   For example, if the equation is given as $5x^2 + 7 = 9x$, rewrite it as $5x^2 - 9x + 7 = 0$. This rearrangement makes it clear what the values of a, b, and c are, setting the stage for the next steps.

2. Identify the Coefficients a, b, and c: Once the equation is in standard form, identify the coefficients a, b, and c. The coefficient a is the number that multiplies $x^2$, b is the number that multiplies x, and c is the constant term. Correctly identifying these coefficients is crucial, as any error here will affect the discriminant calculation and, consequently, the determination of the number of real solutions.

   Using the example $5x^2 - 9x + 7 = 0$, we identify $a = 5$, $b = -9$, and $c = 7$. Pay close attention to the signs of the coefficients, especially b, as they are critical in the discriminant calculation.

3. Calculate the Discriminant: The discriminant is calculated using the formula $D = b^2 - 4ac$. Substitute the values of a, b, and c that you identified in the previous step into this formula. Performing the calculation accurately is vital, as the value of the discriminant will determine the nature of the solutions.

   For our example, we calculate the discriminant as follows: $D = (-9)^2 - 4(5)(7)$ $D = 81 - 140$ $D = -59$

4. Determine the Number of Real Solutions Based on the Discriminant's Value: The value of the discriminant provides direct insight into the number of real solutions the quadratic equation has. Here’s how to interpret the discriminant:

   • If $D > 0$: The equation has two distinct real solutions. This means the graph of the quadratic equation (a parabola) intersects the x-axis at two different points.
   • If $D = 0$: The equation has exactly one real solution, which is also known as a repeated or double root. In this case, the parabola touches the x-axis at exactly one point.
   • If $D < 0$: The equation has no real solutions. Instead, it has two complex solutions. The parabola does not intersect the x-axis.

   In our example, the discriminant $D = -59$ is less than 0. Therefore, the quadratic equation $5x^2 - 9x + 7 = 0$ has no real solutions.

By following these steps carefully, you can confidently use the discriminant to determine the nature and number of real solutions for any quadratic equation.

Examples of Discriminant Use

To further illustrate how the discriminant is used to determine the number of real solutions, let's explore a few more examples:

Example 1: Two Distinct Real Solutions

Consider the quadratic equation $x^2 - 5x + 6 = 0$. Identify the coefficients:

• $a = 1$
• $b = -5$
• $c = 6$

Calculate the discriminant:

$D = (-5)^2 - 4(1)(6)$ $D = 25 - 24$ $D = 1$

Since $D = 1 > 0$, the equation has two distinct real solutions.

Example 2: One Real Solution (Repeated Root)

Consider the quadratic equation $4x^2 + 4x + 1 = 0$. Identify the coefficients:

• $a = 4$
• $b = 4$
• $c = 1$

Calculate the discriminant:

$D = (4)^2 - 4(4)(1)$ $D = 16 - 16$ $D = 0$

Since $D = 0$, the equation has one real solution (a repeated root).

Example 3: No Real Solutions

Consider the quadratic equation $2x^2 + 3x + 5 = 0$. Identify the coefficients:

• $a = 2$
• $b = 3$
• $c = 5$

Calculate the discriminant:

$D = (3)^2 - 4(2)(5)$ $D = 9 - 40$ $D = -31$

Since $D = -31 < 0$, the equation has no real solutions.

These examples illustrate the practical application of the discriminant in determining the nature and number of solutions for various quadratic equations. By calculating the discriminant, we can quickly ascertain whether the equation has two distinct real solutions, one real solution (repeated root), or no real solutions, which simplifies the process of analyzing quadratic equations.

Importance of Understanding the Discriminant

Understanding the discriminant is crucial in the study of quadratic equations for several reasons. Firstly, it offers a quick and efficient way to determine the nature of the solutions without having to solve the entire equation. This can save a significant amount of time, especially in situations where only the type of solutions is required, not the solutions themselves. For example, in applied problems where the existence of a real solution has a physical meaning, knowing the discriminant can provide immediate answers.

Secondly, the discriminant provides valuable insights into the graphical representation of quadratic equations. Since a quadratic equation forms a parabola when graphed, the discriminant tells us how the parabola intersects the x-axis. A positive discriminant indicates that the parabola intersects the x-axis at two distinct points, a discriminant of zero indicates that the parabola touches the x-axis at one point (the vertex), and a negative discriminant indicates that the parabola does not intersect the x-axis at all. This graphical interpretation enhances the understanding of quadratic functions and their behavior.

Moreover, the discriminant is a fundamental concept that extends beyond basic algebra. It appears in more advanced mathematical contexts, such as in the study of conic sections, polynomial equations of higher degrees, and even in calculus. A solid grasp of the discriminant is therefore essential for students progressing in their mathematical education.

In practical applications, the discriminant can be used in various fields, including physics, engineering, and computer science. For instance, in physics, when analyzing projectile motion, the discriminant can determine whether a projectile will reach a certain height. In engineering, it can be used to assess the stability of systems. In computer science, it can be applied in optimization problems and algorithm design.

In summary, the discriminant is not just a formula; it is a powerful tool that provides deep insights into the nature of quadratic equations and their solutions. Its understanding is crucial for both theoretical mathematics and practical applications, making it an indispensable concept in mathematical education.

Conclusion

The discriminant is a powerful tool in determining the nature and number of real solutions for quadratic equations. By calculating $D = b^2 - 4ac$, we can quickly ascertain whether an equation has two distinct real solutions, one real solution, or no real solutions. This method is not only efficient but also provides a deeper understanding of the behavior of quadratic equations. In the case of the equation $16x^2 + 4 = 16x$, we found that the discriminant is zero, indicating that there is exactly one real solution. Understanding and applying the discriminant is a valuable skill in mathematics, enabling us to analyze quadratic equations effectively and efficiently.