Finding The Perfect 'c': When A Quadratic Equation Has One Solution

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Hey everyone! Today, we're diving into a cool math problem: figuring out what value of 'c' makes the quadratic equation x² - 8x + c = 0 have exactly one solution. Sounds tricky, right? Don't worry, we'll break it down step by step, making it super easy to understand. This is a classic example of how the discriminant of a quadratic equation helps us determine the nature of its roots. Understanding this concept is key to solving a wide range of problems in algebra. Let's get started!

Understanding Quadratic Equations and Solutions

First off, let's refresh our memory about quadratic equations. A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solutions to a quadratic equation are the values of 'x' that satisfy the equation. These solutions are often called the roots or zeros of the equation. A quadratic equation can have:

  • Two distinct real solutions: This happens when the graph of the equation (a parabola) crosses the x-axis at two different points.
  • One real solution (a repeated root): This occurs when the parabola touches the x-axis at exactly one point. It's like the vertex of the parabola is sitting right on the x-axis.
  • No real solutions: This means the parabola doesn't intersect the x-axis at all; it either floats above or below it. In this case, the solutions are complex numbers.

In our specific problem, we're looking for the value of 'c' that results in the equation x² - 8x + c = 0 having only one solution. This means we're looking for a parabola that barely kisses the x-axis. Think of it like a perfectly balanced equation where the parabola's lowest (or highest, depending on the coefficient of x²) point just touches the x-axis.

To really grasp this, consider the graphical representation of a quadratic equation. The equation x² - 8x + c = 0 represents a parabola. The solutions to this equation are the x-intercepts of the parabola, the points where the parabola crosses or touches the x-axis. When a quadratic equation has exactly one solution, the parabola touches the x-axis at a single point, which is the vertex of the parabola. This is the key concept we need to understand to solve the problem. Therefore, finding the value of 'c' involves manipulating the equation so that the parabola's vertex lies on the x-axis.

The Discriminant: Our Secret Weapon

Okay, now let's introduce our secret weapon: the discriminant. The discriminant is a part of the quadratic formula and is denoted as Δ (Delta). It's calculated using the formula: Δ = b² - 4ac. The discriminant helps us determine how many real solutions a quadratic equation has. Here’s the breakdown:

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

Since we want our equation x² - 8x + c = 0 to have exactly one solution, we need the discriminant to be equal to zero. In our equation, a = 1, b = -8, and c = c (the value we're trying to find). So, let's plug these values into the discriminant formula: Δ = (-8)² - 4(1)(c). Simplifying this gives us Δ = 64 - 4c.

We want Δ = 0 because a single solution indicates a repeated root, and the discriminant is a tool for determining the nature of roots. Setting the discriminant to zero allows us to pinpoint the specific condition for x² - 8x + c = 0 to have one solution. This is not just a mathematical trick; it's a direct application of the theory that the discriminant reflects the nature of the solutions to a quadratic equation. By setting the discriminant equal to zero, we are essentially finding the specific value of 'c' that makes the quadratic equation perfectly balanced, resulting in the parabola just touching the x-axis at its vertex.

Solving for 'c': The Grand Finale

Now we're ready for the grand finale – finding the value of 'c'! We know that for our equation to have one solution, the discriminant must be zero. So, let's set our discriminant equation to zero and solve for 'c':

  • 64 - 4c = 0

Now, let’s solve this simple linear equation:

  1. Subtract 64 from both sides: -4c = -64
  2. Divide both sides by -4: c = 16

And there you have it, guys! When c = 16, the quadratic equation x² - 8x + 16 = 0 has exactly one solution.

To confirm, we can substitute c = 16 back into our original equation, resulting in x² - 8x + 16 = 0. Factoring this equation, we get (x - 4)² = 0, which gives us the single solution x = 4. This confirms that our calculation is correct, and we have successfully found the value of 'c' that yields only one solution. The solution x=4 represents the x-coordinate where the parabola touches the x-axis. At this point, the equation has no other real roots, hence the single solution we were seeking. This result perfectly aligns with the graphical interpretation we discussed earlier.

Conclusion: Wrapping It Up

So, to recap, for the quadratic equation x² - 8x + c = 0 to have exactly one solution, the value of 'c' must be 16. This is because, at c = 16, the discriminant becomes zero, indicating a single, repeated root. We’ve learned how to use the discriminant to determine the number of solutions a quadratic equation has, which is a super valuable skill in algebra. Keep practicing, and you'll become a pro at these types of problems!

This method can be applied to many similar problems. The key is understanding the relationship between the discriminant and the number of solutions. The same approach can be used to determine the value of 'c' for which a quadratic equation has two distinct solutions (when Δ > 0) or no real solutions (when Δ < 0). Remember, the core concept behind all of these is the discriminant, which acts as a guide to the nature of the roots of any quadratic equation. In summary, we utilized the discriminant to understand the types of solutions a quadratic equation possesses. It has allowed us to not only solve for 'c' but also deepen our understanding of quadratic equations, thereby strengthening our mathematical toolkit.