Solving Quadratic Equations Finding The Other Solution For X² - 2x - 15 = 0

Hey there, math enthusiasts! Today, we're diving into the fascinating world of quadratic equations, those intriguing expressions that often hold the key to solving real-world problems. We've got a specific equation on our hands: x² - 2x - 15 = 0. One solution is already revealed to us: x = -3. But the puzzle isn't complete yet! Our mission is to uncover the other solution, the hidden twin that satisfies this equation. Think of it as a treasure hunt, where each step brings us closer to the prize – the elusive value of x.

Cracking the Code: Methods to Solve Quadratic Equations

Before we jump into solving our particular equation, let's equip ourselves with the tools we need. There are several powerful methods for tackling quadratic equations, each with its own strengths and nuances. We'll explore a few popular techniques, giving you a well-rounded arsenal for conquering these mathematical challenges.

Factoring: The Art of Decomposition

Factoring is like being a mathematical detective, breaking down a complex expression into simpler components. For a quadratic equation in the form ax² + bx + c = 0, factoring involves finding two numbers that multiply to give 'c' and add up to 'b'. These numbers then help us rewrite the quadratic expression as a product of two binomials. When we can successfully factor, we unlock the solutions almost effortlessly.

In our case, x² - 2x - 15 = 0, we need to find two numbers that multiply to -15 and add up to -2. Take a moment to ponder the possibilities. Which numbers fit this description? With a little thought, you'll discover that -5 and 3 are the perfect match! They multiply to -15 and add up to -2. This allows us to factor the equation as (x - 5)(x + 3) = 0. Now, the magic happens. For the product of two factors to be zero, at least one of them must be zero. So, either x - 5 = 0 or x + 3 = 0. Solving these simple equations gives us x = 5 and x = -3. Voila! We've found both solutions, using the power of factoring. This method is particularly elegant when the quadratic expression is easily factorable, offering a quick and satisfying path to the answer.

The Quadratic Formula: A Universal Key

For those equations that stubbornly resist factoring, we have a trusty ally: the quadratic formula. This formula, a cornerstone of algebra, provides a guaranteed solution for any quadratic equation, regardless of its factorability. It might look a bit intimidating at first glance, but with practice, it becomes a familiar friend. The quadratic formula states that for an equation ax² + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / 2a

Let's break this down. 'a', 'b', and 'c' are the coefficients of our quadratic equation. The symbol '±' means we have two possible solutions, one with a plus sign and one with a minus sign. The square root part, √(b² - 4ac), is called the discriminant, and it reveals the nature of the solutions. A positive discriminant indicates two real solutions, a zero discriminant indicates one real solution (a repeated root), and a negative discriminant indicates two complex solutions.

Applying the quadratic formula to our equation, x² - 2x - 15 = 0, we identify a = 1, b = -2, and c = -15. Plugging these values into the formula, we get:

x = (2 ± √((-2)² - 4 * 1 * -15)) / (2 * 1) x = (2 ± √(4 + 60)) / 2 x = (2 ± √64) / 2 x = (2 ± 8) / 2

This gives us two solutions:

x = (2 + 8) / 2 = 5 x = (2 - 8) / 2 = -3

Just like with factoring, we've arrived at the solutions x = 5 and x = -3. The quadratic formula may involve a bit more computation, but it's a reliable workhorse that never fails to deliver the goods. It's the go-to method when factoring seems like a dead end, ensuring that we can always crack the code of a quadratic equation.

Completing the Square: A Step-by-Step Transformation

Completing the square is another powerful technique for solving quadratic equations, and it offers valuable insights into the structure of these equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. This method might seem a bit more involved than factoring or the quadratic formula, but it's a great way to deepen your understanding of quadratic equations and their properties.

The basic idea behind completing the square is to transform the quadratic expression into the form (x + p)² + q, where p and q are constants. This form reveals the vertex of the parabola represented by the quadratic equation, which is a key feature of its graph.

Let's apply completing the square to our equation, x² - 2x - 15 = 0. The first step is to move the constant term to the right side of the equation:

x² - 2x = 15

Now, we focus on the left side, x² - 2x. To complete the square, we need to add a constant term that will make this expression a perfect square trinomial. This constant is found by taking half of the coefficient of the x term (which is -2), squaring it ((-1)² = 1), and adding it to both sides of the equation:

x² - 2x + 1 = 15 + 1

x² - 2x + 1 = 16

Now, the left side is a perfect square trinomial, which can be factored as (x - 1)²:

(x - 1)² = 16

Taking the square root of both sides, we get:

x - 1 = ±4

This gives us two equations:

x - 1 = 4 or x - 1 = -4

Solving for x, we find:

x = 5 or x = -3

Once again, we've arrived at the solutions x = 5 and x = -3. Completing the square might take a few more steps, but it's a valuable technique that reinforces our understanding of quadratic equations and their connections to parabolas. It's a powerful tool to have in your mathematical toolkit.

Unveiling the Second Solution: Putting Our Knowledge to the Test

Now that we've explored various methods for solving quadratic equations, let's circle back to our original problem: x² - 2x - 15 = 0. We already know that one solution is x = -3. Our goal is to find the other solution. We can leverage the techniques we've learned to crack this puzzle.

Remember our factoring adventure? We discovered that x² - 2x - 15 can be factored as (x - 5)(x + 3). This immediately reveals the solutions: x = 5 and x = -3. So, the other solution we've been searching for is x = 5.

Alternatively, we could use the quadratic formula. Plugging in the coefficients a = 1, b = -2, and c = -15, we get the same solutions: x = 5 and x = -3. This confirms our factoring result and demonstrates the power of the quadratic formula as a universal solver.

Completing the square would also lead us to the same destination. By transforming the equation into the form (x - 1)² = 16, we can easily extract the solutions x = 5 and x = -3.

No matter which path we choose, the answer remains the same: the other solution to x² - 2x - 15 = 0 is x = 5. We've successfully navigated the world of quadratic equations and emerged victorious!

Why This Matters: The Power of Quadratic Equations

You might be wondering,