Solving 25x² + 10x + 10: A Step-by-Step Guide

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Solving 25x² + 10x + 10: A Step-by-Step Guide

Hey guys! Today, we're going to dive into solving the quadratic expression 25x² + 10x + 10. This might seem a bit daunting at first, but trust me, we'll break it down step by step so it’s super easy to understand. Quadratic expressions are fundamental in algebra, and mastering them opens doors to solving more complex problems in math and even real-world scenarios. So, let's roll up our sleeves and get started!

Understanding Quadratic Expressions

Before we jump into the solution, let’s quickly recap what a quadratic expression actually is. Quadratic expressions are polynomials of degree two, which means the highest power of the variable (in our case, x) is 2. The general form of a quadratic expression is ax² + bx + c, where a, b, and c are constants. In our expression, 25x² + 10x + 10, a is 25, b is 10, and c is also 10. Identifying these coefficients is the first step in solving the expression.

Now, you might be wondering, “Why do we even care about quadratic expressions?” Well, they pop up everywhere! From calculating the trajectory of a ball thrown in the air to designing the curves of a bridge, quadratic equations are incredibly versatile. Solving them allows us to find the roots or zeros of the expression, which are the values of x that make the expression equal to zero. These roots often represent critical points in real-world applications.

There are several methods to solve quadratic expressions, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and is suitable for different types of quadratic expressions. We’ll explore the most efficient methods for our specific expression, 25x² + 10x + 10, ensuring you have a solid grasp of the process. Remember, the goal here is not just to get the answer but to understand the why behind each step. So, let's get our hands dirty and start solving!

Method 1: Using the Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations of the form ax² + bx + c = 0. It's especially useful when factoring isn't straightforward. The formula is given by:

x = [-b ± √(b² - 4ac)] / (2a)

It might look intimidating, but it’s actually quite simple to use once you get the hang of it. Let’s apply it to our expression, 25x² + 10x + 10. Remember, here a = 25, b = 10, and c = 10.

  1. Plug in the values: Substitute a, b, and c into the formula: x = [-10 ± √(10² - 4 * 25 * 10)] / (2 * 25)

  2. Simplify the expression: Let's break it down step by step:

    • Calculate the discriminant (the part under the square root): b² - 4ac = 10² - 4 * 25 * 10 = 100 - 1000 = -900
    • Substitute the discriminant back into the formula: x = [-10 ± √(-900)] / 50

  3. Handle the imaginary unit: Since the discriminant is negative, we have a square root of a negative number, which introduces the imaginary unit i, where i = √(-1). So, √(-900) = √(900 * -1) = √(900) * √(-1) = 30i

  4. Final solution: Substitute the imaginary component back into the formula: x = [-10 ± 30i] / 50

  5. Simplify further: Divide both the real and imaginary parts by the greatest common divisor, which is 10: x = (-1 ± 3i) / 5

So, the solutions are x = (-1 + 3i) / 5 and x = (-1 - 3i) / 5. These are complex roots, which means the quadratic expression does not intersect the x-axis on a graph. The quadratic formula is a powerful tool, especially when dealing with complex roots or when factoring seems too complicated. It provides a straightforward method to find the solutions, regardless of the nature of the roots.

Method 2: Completing the Square

Completing the square is another technique for solving quadratic equations. It involves transforming the quadratic expression into a perfect square trinomial plus a constant. This method is particularly useful for understanding the structure of quadratic equations and can be applied even when the quadratic formula might seem cumbersome. Let’s see how it works for our expression, 25x² + 10x + 10.

  1. Divide by the leading coefficient: To make the coefficient of x² equal to 1, divide the entire expression by 25: x² + (10/25)x + (10/25) = x² + (2/5)x + (2/5)

  2. Move the constant term: Move the constant term (2/5) to the right side of the equation (we're setting the expression equal to zero for this method): x² + (2/5)x = -2/5

  3. Complete the square: To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of x is 2/5, so half of it is (2/5) / 2 = 1/5, and the square of 1/5 is (1/5)² = 1/25. Add 1/25 to both sides: x² + (2/5)x + 1/25 = -2/5 + 1/25

  4. Factor and simplify: The left side is now a perfect square trinomial, which can be factored as (x + 1/5)² . Simplify the right side: (x + 1/5)² = -10/25 + 1/25 = -9/25

  5. Take the square root: Take the square root of both sides: x + 1/5 = ±√(-9/25) = ±(3i/5)

  6. Solve for x: Subtract 1/5 from both sides: x = -1/5 ± (3i/5)

So, the solutions are x = (-1 + 3i) / 5 and x = (-1 - 3i) / 5, which are the same complex roots we found using the quadratic formula. Completing the square can be a bit more involved, but it offers a deeper understanding of quadratic expressions and is a valuable tool in your mathematical toolkit. It's especially useful in contexts where you need to rewrite the quadratic expression in vertex form, which gives you the vertex of the parabola.

Factoring (When Possible)

While the quadratic formula and completing the square are powerful methods, factoring is often the quickest way to solve a quadratic equation if it's possible. Factoring involves breaking down the quadratic expression into two binomials. However, not all quadratic expressions can be factored easily, especially when the roots are complex or irrational. Let's see if we can factor our expression, 25x² + 10x + 10.

  1. Look for common factors: First, check if there's a common factor that can be factored out from all terms. In our case, we can factor out 5: 5(5x² + 2x + 2)

  2. Attempt to factor the quadratic: Now, we need to factor the expression 5x² + 2x + 2. We're looking for two binomials (ax + b)(cx + d) such that:

    • ac = 5
    • ad + bc = 2
    • bd = 2

  3. Analyze the possibilities: The factors of 5 are 1 and 5, and the factors of 2 are 1 and 2. Trying different combinations, we quickly realize that no integer values of a, b, c, and d will satisfy the conditions. This indicates that the quadratic expression 5x² + 2x + 2 cannot be factored easily using integers.

Since we couldn't find integer factors, this method isn't the most efficient for this particular expression. Factoring works best when the quadratic expression has integer roots. In cases like this, the quadratic formula or completing the square are more reliable methods. Factoring is a fantastic skill to have, but it’s also important to recognize when another method might be more suitable. Always start by looking for common factors and then attempt to factor the quadratic, but be ready to switch to another method if it becomes too cumbersome.

Real-World Applications

Understanding how to solve quadratic expressions isn't just about acing your math exams; it's also about equipping yourself with tools to tackle real-world problems. Quadratic equations pop up in various fields, from physics and engineering to economics and computer science. Let's look at a couple of examples to see how this knowledge can be applied.

  1. Physics – Projectile Motion: One of the most classic applications of quadratic equations is in describing the trajectory of a projectile, like a ball thrown in the air. The height of the ball at any given time can be modeled by a quadratic equation, where the coefficients are determined by the initial velocity, launch angle, and gravitational acceleration. Solving the quadratic equation allows us to find the maximum height the ball reaches or the time it takes to hit the ground. This is crucial in sports science, where athletes and coaches analyze trajectories to optimize performance.

  2. Engineering – Bridge Design: Quadratic equations are also essential in structural engineering, particularly in the design of bridges. The shape of a suspension bridge cable, for instance, often follows a parabolic curve, which can be described by a quadratic equation. Engineers use these equations to calculate the forces acting on the bridge and ensure its stability and safety. Understanding quadratic expressions helps them determine the optimal shape and materials for the bridge, balancing structural integrity with cost-effectiveness.

  3. Economics – Profit Maximization: In business and economics, quadratic functions can model profit, revenue, and cost. For example, the profit function of a company might be represented as a quadratic equation, where the roots indicate break-even points and the vertex represents the point of maximum profit. By solving the quadratic equation, businesses can determine the optimal production level or pricing strategy to maximize their profits. This application highlights the practical value of quadratic equations in decision-making processes.

  4. Computer Science – Algorithm Analysis: In computer science, quadratic equations can appear in the analysis of algorithms. For example, the time complexity of certain algorithms might be described by a quadratic function. Understanding how to solve these equations helps computer scientists optimize algorithms and improve the efficiency of software. This is particularly important in fields like data science and artificial intelligence, where algorithms need to process large amounts of data quickly.

These examples are just the tip of the iceberg. The ability to solve quadratic expressions is a valuable skill that opens doors to a wide range of applications. Whether you're calculating the trajectory of a rocket or optimizing a business strategy, the principles you learn in algebra can have a significant impact in the real world. So, keep practicing and exploring, and you'll be amazed at how versatile quadratic equations can be.

Conclusion

Alright, guys, we've covered a lot in this guide! We tackled the quadratic expression 25x² + 10x + 10 using multiple methods, including the quadratic formula, completing the square, and briefly touched on factoring. Each method has its strengths, and understanding them gives you a versatile toolkit for solving quadratic equations. We found that our expression has complex roots, which means it doesn't intersect the x-axis, and we saw how these roots can be found using both the quadratic formula and completing the square.

Remember, the key to mastering these concepts is practice. The more you work with quadratic expressions, the more comfortable you'll become with identifying the best method for solving them. Don't be afraid to try different approaches and make mistakes – that's how we learn! And, as we discussed, these skills aren't just for the classroom. Quadratic equations are powerful tools that can be applied in various real-world scenarios, from physics and engineering to economics and computer science.

So, keep up the great work, and don't hesitate to revisit these methods whenever you encounter a quadratic expression. Whether you're calculating projectile motion, designing a bridge, or optimizing a business strategy, the ability to solve quadratic equations will be a valuable asset. Keep practicing, stay curious, and you'll be solving quadratic expressions like a pro in no time! If you have any questions or want to explore more examples, feel free to dive deeper into these methods and their applications. Happy solving!