Solving For C In System Of Equations With Two Real Solutions

In this article, we delve into the fascinating world of systems of equations, specifically focusing on a system comprising a linear equation and a quadratic equation. Our primary objective is to determine the possible values of the constant c for which the system has two distinct real solutions. The given system is defined by the equations y = x - c and y = -4(x - 6)^2. This problem elegantly combines algebraic manipulation with geometric intuition, offering a rich exploration of the interplay between linear and quadratic functions. Understanding the conditions under which such a system yields two distinct solutions is crucial in various fields, including engineering, physics, and economics, where modeling real-world phenomena often involves analyzing the intersection of curves and lines.

To effectively tackle this problem, we first need to deeply understand the nature of the equations involved. The first equation, y = x - c, represents a linear function with a slope of 1 and a y-intercept of -c. This means it's a straight line that rises at a 45-degree angle with respect to the x-axis. The constant c dictates the vertical position of this line; changing c shifts the line up or down without altering its slope. The second equation, y = -4(x - 6)^2, is a quadratic function in vertex form. It represents a parabola that opens downwards due to the negative coefficient (-4) in front of the squared term. The vertex of this parabola is at the point (6, 0), and the parabola is stretched vertically by a factor of 4. The solutions to the system of equations correspond to the points where the line and the parabola intersect. For the system to have two distinct real solutions, the line must intersect the parabola at two distinct points. This geometric perspective provides a valuable way to visualize the problem and understand the role of the constant c in determining the number of solutions.

Setting up the Equations for Solving

To find the values of c that result in two distinct real solutions, we need to combine the two equations and analyze the resulting expression. Since both equations are solved for y, we can set them equal to each other: x - c = -4(x - 6)^2. This equation represents the x-coordinates of the intersection points between the line and the parabola. To solve this equation, we first expand the quadratic term: x - c = -4(x^2 - 12x + 36). Then, we distribute the -4: x - c = -4x^2 + 48x - 144. Next, we rearrange the equation into a standard quadratic form by moving all terms to one side: 4x^2 - 47x + (144 - c) = 0. This quadratic equation in x will have solutions corresponding to the x-coordinates of the intersection points. The number of real solutions to this quadratic equation depends on the discriminant, which is a critical concept in determining the nature of the roots of a quadratic equation. Understanding this setup is crucial as it lays the foundation for using the discriminant to find the desired values of c.

Utilizing the Discriminant for Solution Analysis

The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the formula Δ = b^2 - 4ac. This discriminant plays a pivotal role in determining the nature of the roots of the quadratic equation. Specifically: If Δ > 0, the quadratic equation has two distinct real roots. If Δ = 0, the quadratic equation has one real root (a repeated root). If Δ < 0, the quadratic equation has no real roots (two complex roots). In our case, the quadratic equation is 4x^2 - 47x + (144 - c) = 0, so a = 4, b = -47, and c = 144 - c (note that here 'c' is the constant term of the quadratic, not the 'c' from the original linear equation, but to avoid confusion we will proceed using the same notation, making the distinction clear). For our system to have two distinct real solutions, the discriminant must be greater than 0. Thus, we need to calculate the discriminant and set up the inequality: Δ = (-47)^2 - 4(4)(144 - c) > 0. This inequality will help us determine the range of values for c that satisfy the condition of having two distinct real solutions. Accurately calculating and interpreting the discriminant is essential for solving the problem.

Now, let's calculate the discriminant and solve the inequality. We have Δ = (-47)^2 - 4(4)(144 - c) > 0. First, we compute the square: (-47)^2 = 2209. Then, we simplify the expression: 2209 - 16(144 - c) > 0. Next, we distribute the -16: 2209 - 2304 + 16c > 0. Combining the constants, we get: -95 + 16c > 0. Now, we isolate c by adding 95 to both sides: 16c > 95. Finally, we divide by 16: c > 95/16. This inequality tells us that the value of c must be greater than 95/16 for the system of equations to have two distinct real solutions. This result is a crucial finding, as it allows us to evaluate the given options and determine which values of c satisfy the condition. Understanding the algebraic steps involved in solving this inequality is paramount for arriving at the correct conclusion.

Evaluating the Given Options

We have determined that c > 95/16. Now, let's evaluate the given options to see which values of c satisfy this condition. First, we need to approximate the value of 95/16. Dividing 95 by 16, we get approximately 5.9375. So, we are looking for values of c that are greater than 5.9375. The options provided are: A) 1, B) 5, C) 95/16, and D) 11. A) 1 is clearly less than 5.9375, so it does not satisfy the condition. B) 5 is also less than 5.9375, so it does not satisfy the condition. C) 95/16 is equal to 5.9375, but we need c to be strictly greater than 95/16, so it does not satisfy the condition. D) 11 is greater than 5.9375, so it satisfies the condition. Therefore, the only option that could be the value of c is 11. This step demonstrates how the algebraic solution translates into a concrete answer when applied to specific choices.

In conclusion, by analyzing the system of equations y = x - c and y = -4(x - 6)^2, we found that the condition for the system to have two distinct real solutions is c > 95/16. This was achieved by setting the equations equal to each other, rearranging them into a quadratic form, and applying the discriminant condition for two distinct real roots. After calculating the discriminant and solving the inequality, we evaluated the given options and determined that only c = 11 satisfies the condition. This problem exemplifies how algebraic techniques, combined with a solid understanding of quadratic equations and the discriminant, can be used to solve complex mathematical problems. The process highlights the importance of translating a geometric problem (intersection of curves) into an algebraic one (solving a quadratic equation) and using the discriminant as a powerful tool for analysis. Understanding these concepts is crucial for success in algebra and related fields.

Given the system of equations: y = x - c and y = -4(x - 6)^2, where c is a constant, and the system has two distinct real solutions, which of the following could be the value of c? Options: A) 1, B) 5, C) 95/16, D) 11.

Solving for c in a System of Equations with Two Distinct Real Solutions