Largest 'n' For Unique Solution: Quadratic Equation Solver

Hey guys! Ever wondered how to find the largest possible value of a parameter that makes a quadratic equation have only one solution? It's a pretty cool problem that combines algebra and a bit of clever thinking. Let's dive into a specific example and break it down step by step. We're going to tackle the equation $x^2 + 2(n-3)x + 4n = 0$ and figure out the biggest '$n$' can be while still giving us just one unique solution for '$x$'.

Understanding the Discriminant and Unique Solutions

First off, the key to solving this lies in understanding the discriminant of a quadratic equation. You might remember that the discriminant (often written as the Greek letter delta, Δ, or simply as 'D') is the part of the quadratic formula under the square root: $b^2 - 4ac$. This little expression tells us a lot about the solutions (also called roots) of our quadratic equation, which is in the general form $ax^2 + bx + c = 0$.

• 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 (it has two complex solutions).

Since we want our equation to have a unique solution, we need the discriminant to be equal to zero. This is the golden rule for solving this type of problem. Our main focus on discriminant allows us to find the condition for a unique solution.

Applying the Discriminant to Our Equation

Alright, let's apply this to our specific equation: $x^2 + 2(n-3)x + 4n = 0$

Here, we can identify:

• a = 1 (the coefficient of $x^2$)
• b = 2(n - 3) (the coefficient of $x$)
• c = 4n (the constant term)

Now, let's plug these values into the discriminant formula and set it equal to zero:

$$D = b^2 - 4ac = [2(n-3)]^2 - 4(1)(4n) = 0$$

Time to do some algebra! First, we square the term 2(n-3):

$$[2(n-3)]^2 = 4(n-3)^2 = 4(n^2 - 6n + 9) = 4n^2 - 24n + 36$$

Next, we calculate 4ac:

$$4(1)(4n) = 16n$$

Now, let's put it all together:

$$4n^2 - 24n + 36 - 16n = 0$$

Combine those 'n' terms to simplify:

$$4n^2 - 40n + 36 = 0$$

Solving the Quadratic for 'n'

We've got a new quadratic equation, but this time it's in terms of 'n'! To make our lives easier, let's divide the entire equation by 4:

$$n^2 - 10n + 9 = 0$$

Now we need to solve for 'n'. We can do this by factoring the quadratic. We're looking for two numbers that multiply to 9 and add up to -10. Those numbers are -1 and -9. So, we can factor the equation as:

$$(n - 1)(n - 9) = 0$$

This gives us two possible solutions for 'n':

• n = 1
• n = 9

Identifying the Largest Value of 'n'

The question asks for the largest value of 'n'. Comparing our two solutions, it's clear that:

$$n = 9$$

is the larger value. So, the largest value of '$n$' for which the original equation has a unique solution is 9. It's essential to identify the largest value to fulfill the question's requirement.

Checking Our Answer (Always a Good Idea!)

To be extra sure, let's plug n = 9 back into the original equation and see what happens:

$$x^2 + 2(9-3)x + 4(9) = 0$$

$$x^2 + 12x + 36 = 0$$

This equation can be factored as:

$$(x + 6)(x + 6) = 0$$

$$(x + 6)^2 = 0$$

Which gives us a single solution:

$$x = -6$$

This confirms that when n = 9, the original equation has only one solution. We can perform this verification to ensure the correctness of our solution.

Key Takeaways and Why This Matters

• The discriminant is your friend: Whenever you're dealing with quadratic equations and want to know about the nature of the solutions, the discriminant is your go-to tool. Remember D = $b^2 - 4ac$!
• Unique solutions mean D = 0: This is the core concept we used to solve the problem. If a quadratic has only one solution, the discriminant must be zero.
• Factoring and solving quadratics are essential skills: Being comfortable with factoring or using the quadratic formula is crucial for solving these types of problems. The ability to factor quadratic equations is a fundamental skill in algebra.
• Always check your answer: Plugging your solution back into the original equation is a great way to catch mistakes and build confidence in your answer.

Variations and Extensions

This type of problem can be varied in a few ways. For example:

• Finding the range of 'n' for two distinct solutions: You'd set D > 0 and solve the resulting inequality.
• Finding the values of 'n' for no real solutions: You'd set D < 0 and solve the inequality.
• Using different types of equations: The same principle of the discriminant can be applied to other types of equations that can be reduced to a quadratic form.

Understanding how the discriminant works opens up a whole world of problem-solving possibilities!

Conclusion

So, there you have it! We successfully calculated the largest value of '$n$' for which the given quadratic equation has a unique solution. The key was understanding the discriminant and how it relates to the number of solutions. Remember, guys, math isn't just about formulas; it's about understanding the concepts and applying them creatively. Keep practicing, and you'll become a quadratic equation whiz in no time! The importance of practice cannot be overstated in mastering mathematical concepts.