Discriminant Of 9x² + 2 = 10x How To Calculate
In the realm of mathematics, specifically when dealing with quadratic equations, the discriminant plays a pivotal role in determining the nature and number of solutions. When we encounter a quadratic equation in the standard form of ax² + bx + c = 0, the discriminant, often denoted as Δ (delta), is given by the formula Δ = b² - 4ac. This seemingly simple formula holds the key to unlocking crucial information about the roots (or solutions) of the quadratic equation. Let's delve deeper into the significance of the discriminant and its applications.
The discriminant acts as a powerful indicator, revealing whether a quadratic equation has two distinct real roots, one repeated real root, or two complex roots. The value of the discriminant dictates the characteristics of the solutions. When Δ > 0, the quadratic equation possesses two distinct real roots, indicating that the parabola intersects the x-axis at two different points. This scenario signifies that there are two unique values of x that satisfy the equation. Conversely, if Δ = 0, the equation has exactly one real root, which is a repeated root. In this case, the parabola touches the x-axis at a single point, representing a single solution. Finally, when Δ < 0, the quadratic equation has two complex roots, implying that the parabola does not intersect the x-axis at all. The roots are complex conjugates, involving imaginary numbers.
To fully grasp the concept, consider the quadratic formula, which provides the solutions to any quadratic equation: x = (-b ± √Δ) / 2a. Notice how the discriminant appears under the square root. This placement is crucial because the square root of a positive number is real, the square root of zero is zero, and the square root of a negative number is imaginary. Thus, the discriminant directly influences the nature of the roots. A positive discriminant leads to two distinct real roots because we are adding and subtracting a real number. A zero discriminant results in a single real root since the square root term vanishes. A negative discriminant introduces imaginary numbers, leading to complex roots.
Now, let's apply this knowledge to the specific equation at hand: 9x² + 2 = 10x. The first crucial step in determining the discriminant is to rewrite the equation in the standard quadratic form, ax² + bx + c = 0. This involves rearranging the terms to bring all terms to one side of the equation, leaving zero on the other side. Subtracting 10x from both sides of the equation, we obtain: 9x² - 10x + 2 = 0. This rearranged equation now clearly displays the coefficients a, b, and c that we need to calculate the discriminant.
In this standard form, we can identify the coefficients as follows: a = 9, b = -10, and c = 2. These values are the key ingredients for calculating the discriminant. Now, we can substitute these values into the discriminant formula, Δ = b² - 4ac. Plugging in the values, we get Δ = (-10)² - 4 * 9 * 2. Evaluating this expression, we find Δ = 100 - 72, which simplifies to Δ = 28. This calculation is a straightforward application of the formula, but it is essential to ensure accuracy to arrive at the correct discriminant value.
The calculated discriminant, Δ = 28, is a positive number. This positive value has significant implications for the nature of the roots of the quadratic equation. As we discussed earlier, a positive discriminant indicates that the quadratic equation has two distinct real roots. This means that the parabola represented by the equation 9x² - 10x + 2 = 0 intersects the x-axis at two different points. These points correspond to the two real solutions of the equation. To find the actual values of these roots, one would typically use the quadratic formula, but the discriminant alone has already provided us with valuable information about the type and number of solutions.
To ensure clarity and understanding, let's break down the calculation of the discriminant for the equation 9x² + 2 = 10x into a step-by-step process. This methodical approach will not only solidify the concept but also prevent potential errors in the calculation.
Step 1: Rewrite the Equation in Standard Form
The initial equation, 9x² + 2 = 10x, is not in the standard quadratic form, ax² + bx + c = 0. To proceed with the discriminant calculation, we must first rearrange the equation into this standard form. This involves moving all terms to one side of the equation, leaving zero on the other side. We achieve this by subtracting 10x from both sides of the equation:
9x² + 2 - 10x = 10x - 10x
This simplifies to:
9x² - 10x + 2 = 0
Now, the equation is in the standard form, and we can clearly identify the coefficients.
Step 2: Identify the Coefficients a, b, and c
In the standard form ax² + bx + c = 0, the coefficients are the numerical values that multiply the variables and the constant term. Comparing our rearranged equation, 9x² - 10x + 2 = 0, with the standard form, we can identify the coefficients as follows:
- a = 9 (the coefficient of x²)
- b = -10 (the coefficient of x)
- c = 2 (the constant term)
These coefficients are the key inputs for the discriminant formula.
Step 3: Apply the Discriminant Formula
The discriminant, denoted as Δ, is calculated using the formula Δ = b² - 4ac. Now that we have identified the coefficients a, b, and c, we can substitute these values into the formula. Plugging in the values, we get:
Δ = (-10)² - 4 * 9 * 2
This is the crucial step where we apply the formula and substitute the correct values. Careful attention to detail is essential here to avoid errors in the calculation.
Step 4: Simplify the Expression
Now, we simplify the expression following the order of operations (PEMDAS/BODMAS). First, we calculate the square:
(-10)² = 100
Next, we perform the multiplication:
4 * 9 * 2 = 72
Now, substitute these values back into the discriminant expression:
Δ = 100 - 72
Finally, perform the subtraction:
Δ = 28
Thus, the discriminant of the quadratic equation 9x² - 10x + 2 = 0 is 28.
Step 5: Interpret the Result
The final step is to interpret the result. We have calculated the discriminant to be Δ = 28. As we know, the value of the discriminant provides information about the nature of the roots of the quadratic equation. Since 28 is a positive number (Δ > 0), we can conclude that the quadratic equation has two distinct real roots. This means that there are two different real numbers that, when substituted for x in the equation, will satisfy the equation. This completes the step-by-step calculation and interpretation of the discriminant.
Having calculated the discriminant for the quadratic equation 9x² - 10x + 2 = 0 and found it to be Δ = 28, a positive value, it's essential to fully understand the implications of this result. A positive discriminant is a significant indicator, revealing crucial information about the nature and number of solutions to the equation.
As established earlier, a positive discriminant (Δ > 0) signifies that the quadratic equation has two distinct real roots. This means there are two different real numbers that satisfy the equation. Geometrically, this corresponds to the parabola represented by the quadratic equation intersecting the x-axis at two distinct points. Each of these points represents a real solution to the equation. The x-coordinates of these intersection points are the roots of the equation.
The presence of two distinct real roots has several practical implications. In various real-world applications, quadratic equations are used to model phenomena such as projectile motion, optimization problems, and curve fitting. In these contexts, the roots of the equation often represent critical values or points of interest. For instance, in projectile motion, the roots might represent the times at which the projectile reaches a certain height or lands on the ground. In optimization problems, the roots may correspond to maximum or minimum values of a function.
To further illustrate, consider the graph of the quadratic equation 9x² - 10x + 2 = 0. Since the discriminant is positive, the parabola intersects the x-axis at two points. These points represent the two real solutions to the equation. If we were to solve the equation using the quadratic formula, we would obtain two different real values for x. These values are the x-coordinates of the intersection points. The positive discriminant assures us that these solutions exist and are real numbers.
The magnitude of the discriminant also provides some insight into the separation between the roots. A larger positive discriminant generally indicates a greater separation between the two real roots. However, the precise values of the roots can only be determined by using the quadratic formula or other solution methods.
In summary, the discriminant is a powerful tool for analyzing quadratic equations. By calculating the discriminant, we can quickly determine the nature and number of solutions without actually solving the equation. For the equation 9x² + 2 = 10x, we rearranged it into standard form, identified the coefficients, applied the discriminant formula, and found Δ = 28. This positive discriminant confirms that the equation has two distinct real roots. This understanding is crucial in various mathematical and real-world applications, allowing us to interpret the solutions in meaningful ways. The discriminant serves as a cornerstone in the study of quadratic equations, providing valuable insights into their behavior and solutions.
