Determine The Number Of Real Solutions For Quadratic Equations

Determining the number of real solutions a quadratic equation possesses is a fundamental concept in algebra. Understanding this concept allows us to analyze and solve a wide range of problems, from simple equations to complex mathematical models. This article delves into the methods for determining the number of real solutions, focusing on the discriminant, and provides step-by-step explanations with examples. We will explore how the discriminant helps us classify quadratic equations into those with two distinct real solutions, one real solution (a repeated root), or no real solutions (complex roots). This understanding is crucial not only for academic purposes but also for various real-world applications in physics, engineering, and economics.

Understanding Quadratic Equations

Before we dive into determining the number of real solutions, let's first recap the standard form of a quadratic equation and its components. A quadratic equation is an equation of the form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a ≠ 0. The solutions to this equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola represented by the quadratic equation intersects the x-axis. The number of these intersection points corresponds to the number of real solutions.

To find the solutions of a quadratic equation, we can use several methods, including factoring, completing the square, and the quadratic formula. The quadratic formula is particularly useful as it provides a direct way to find the solutions regardless of whether the equation can be easily factored. The quadratic formula is given by:

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

In this formula, the expression inside the square root, b² - 4ac, plays a crucial role in determining the nature and number of solutions. This expression is called the discriminant, and we will explore its significance in the next section.

The Discriminant: Key to Real Solutions

The discriminant, denoted as Δ (Delta), is the part of the quadratic formula under the square root:

$Δ = b^2 - 4ac$

The discriminant provides invaluable information about the nature and number of solutions of a quadratic equation without actually solving the equation. By evaluating the discriminant, we can quickly determine whether the quadratic equation has two distinct real solutions, one real solution (a repeated root), or no real solutions (complex roots). This is 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.

Here’s how the discriminant helps us classify the solutions:

1. If Δ > 0: The equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
2. If Δ = 0: The equation has exactly one real solution (a repeated root). This means the parabola touches the x-axis at one point, the vertex of the parabola.
3. If Δ < 0: The equation has no real solutions. The solutions are complex conjugates. This means the parabola does not intersect the x-axis.

Understanding the discriminant is essential for efficiently solving quadratic equations and interpreting their solutions in various contexts. Now, let's apply this knowledge to the given equations.

Analyzing the Equations

Now, let's apply our understanding of the discriminant to determine the number of real solutions for each given quadratic equation. We will calculate the discriminant for each equation and use its value to determine the nature of the solutions.

1. $y = 12x^2 - 9x + 4$

To determine the number of real solutions for the quadratic equation $y = 12x^2 - 9x + 4$, we need to find the discriminant. First, we identify the coefficients a, b, and c. In this case, a = 12, b = -9, and c = 4. Next, we substitute these values into the discriminant formula:

$Δ = b^2 - 4ac = (-9)^2 - 4(12)(4)$

Calculating the discriminant:

$Δ = 81 - 192 = -111$

Since the discriminant Δ = -111, which is less than 0, the equation has no real solutions. This means the parabola represented by the equation does not intersect the x-axis. The solutions are complex conjugates.

Therefore, the equation $y = 12x^2 - 9x + 4$ has zero real solutions.

2. $4y - 7 = 5x^2 - x + 2 + 3y$

To determine the number of real solutions for the equation $4y - 7 = 5x^2 - x + 2 + 3y$, we first need to rewrite the equation in the standard quadratic form, $ax^2 + bx + c = 0$. To do this, we'll rearrange the terms and simplify the equation. We can start by simplifying the left and right sides separately:

Left side: $4y - 7$

Right side: $5x^2 - x + 2 + 3y$

Now, let's move all terms to one side of the equation. We'll subtract $4y$ from both sides:

$-7 = 5x^2 - x + 2 + 3y - 4y$

$-7 = 5x^2 - x + 2 - y$

Next, add $y$ to both sides:

$y - 7 = 5x^2 - x + 2$

Now, isolate $y$ by adding 7 to both sides:

$y = 5x^2 - x + 2 + 7$

$y = 5x^2 - x + 9$

Now that the equation is in the form $y = 5x^2 - x + 9$, we can rewrite it as $5x^2 - x + 9 - y = 0$. However, to analyze it as a quadratic equation in terms of x, we consider the equation $5x^2 - x + (9 - y) = 0$. Here, the coefficients are a = 5, b = -1, and c = 9 (considering y as a constant for now).

Let's calculate the discriminant:

$Δ = b^2 - 4ac = (-1)^2 - 4(5)(9)$

$Δ = 1 - 180 = -179$

Since the discriminant Δ = -179, which is less than 0, the equation $5x^2 - x + 9 = y$ has no real solutions for x. This means for any given value of y, there are no real values of x that satisfy the equation. Graphically, this indicates that the parabola opens upwards and its vertex is above the x-axis, so it does not intersect the x-axis.

Therefore, the equation $4y - 7 = 5x^2 - x + 2 + 3y$ has zero real solutions.

3. $10x + y = 25x^2 + 1$

To determine the number of real solutions for the equation $10x + y = 25x^2 + 1$, we first need to rewrite the equation in the standard quadratic form, $ax^2 + bx + c = 0$. To do this, we rearrange the terms and set the equation equal to zero. Start by moving all terms to one side:

$0 = 25x^2 + 1 - 10x - y$

Rearrange the terms to match the standard form:

$25x^2 - 10x + (1 - y) = 0$

Here, the coefficients are a = 25, b = -10, and c = 1 - y. Now we can calculate the discriminant:

$Δ = b^2 - 4ac = (-10)^2 - 4(25)(1 - y)$

$Δ = 100 - 100(1 - y)$

$Δ = 100 - 100 + 100y$

$Δ = 100y$

The number of real solutions depends on the value of y. Let's analyze the cases:

• If y > 0: The discriminant Δ = 100y > 0, so there are two distinct real solutions.
• If y = 0: The discriminant Δ = 100(0) = 0, so there is exactly one real solution (a repeated root).
• If y < 0: The discriminant Δ = 100y < 0, so there are no real solutions.

Without a specific value for y, we cannot determine a single number of real solutions. However, if we consider the case where we are looking for solutions in x for a fixed y, the number of real solutions varies depending on the value of y. For instance, if y = 1, then Δ = 100, and there are two real solutions. If y = 0, then Δ = 0, and there is one real solution. If y = -1, then Δ = -100, and there are no real solutions.

If the question implies a general case without a specific y, we can analyze the conditions for each number of real solutions:

• Two real solutions: This occurs when 100y > 0, which simplifies to y > 0.
• One real solution: This occurs when 100y = 0, which simplifies to y = 0.
• No real solutions: This occurs when 100y < 0, which simplifies to y < 0.

Since the question does not provide a specific value for y, we cannot give a single numerical answer for the number of real solutions. The number of real solutions depends on the value of y.

Conclusion

In conclusion, determining the number of real solutions for quadratic equations is a critical skill in algebra. By understanding the role of the discriminant (b² - 4ac), we can efficiently classify quadratic equations into those with two distinct real solutions, one real solution, or no real solutions. This analysis not only aids in solving equations but also provides insights into the nature of the solutions and the graphical representation of the quadratic functions. Through the examples provided, we have demonstrated how to apply the discriminant to various quadratic equations and interpret the results. This knowledge is fundamental for further studies in mathematics and its applications in real-world scenarios.