Solving Quadratic Equations: A Step-by-Step Guide

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Hey math enthusiasts! Ready to dive into the world of quadratic equations? This guide is designed to help you solve quadratic equations, covering various techniques and providing clear, step-by-step solutions. We'll tackle the given problems, break down the process, and make sure you've got a solid grasp of this essential math concept. So, buckle up, and let's get started!

Understanding Quadratic Equations

Before we jump into solving equations, let's quickly review what a quadratic equation is. In its standard form, a quadratic equation is written as: $ax^2 + bx + c = 0$ where a, b, and c are constants, and a is not equal to zero. The highest power of the variable (x in this case) is 2, which is why they're called 'quadratic' (quad means 'square'). These equations can have two solutions, one solution, or no real solutions, depending on the values of a, b, and c and the discriminant (we will discuss this later). The solutions to a quadratic equation are also known as its roots or zeros. These are the values of x that satisfy the equation, meaning that when you plug them back into the equation, the equation holds true.

The Importance of Quadratic Equations

Quadratic equations are fundamental in various fields beyond just the classroom. In physics, they're used to model projectile motion, calculate trajectories, and analyze the path of objects under gravity. For example, when you throw a ball, its path follows a parabolic curve described by a quadratic equation. In engineering, they're crucial for designing structures, determining stress points, and optimizing the performance of systems. Architects use these equations to design curved structures and calculate the area of complex shapes. In finance, quadratic equations can model investment growth, predict market trends, and calculate optimal strategies. Even in everyday life, you might indirectly use quadratic equations when estimating distances, calculating areas, or making decisions based on proportional relationships. Understanding these equations opens doors to many practical applications and enhances your problem-solving skills.

Different Methods to Solve Quadratic Equations

There are several methods to solve quadratic equations, each with its advantages depending on the equation's form. The most common methods include:

  1. Factoring: This method involves rewriting the equation in the form of two binomials multiplied together, like (x + p)(x + q) = 0. If you can factor the quadratic equation easily, this method is usually the fastest.
  2. Completing the Square: This is a powerful technique where you manipulate the equation to create a perfect square trinomial on one side. This method is particularly useful when factoring isn't straightforward.
  3. Quadratic Formula: The quadratic formula is a universal method that works for any quadratic equation. It provides a direct solution for x, regardless of whether the equation can be factored. The formula is: $x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where a, b, and c are coefficients from the standard form.

Each method has its place, and choosing the right one can make the solving process much more efficient. For example, factoring is quick if you can spot the factors easily. Completing the square is great when you need to understand the structure of the equation better. And the quadratic formula is the go-to when other methods are cumbersome or when you want a guaranteed solution.

Solving the Quadratic Equation: x=0.5x2x = 0.5x^2

Alright, let's get our hands dirty with the first equation. We need to solve x=0.5x2x = 0.5x^2. To get started, we'll rearrange the equation to the standard form. This means we want to have all the terms on one side and zero on the other.

Step-by-Step Solution

  1. Rearrange the Equation: Subtract x from both sides to get: $0.5x^2 - x = 0$
  2. Multiply by 2 to Remove Decimal: To make the equation easier to work with, let's multiply everything by 2: $x^2 - 2x = 0$
  3. Factor the Equation: Now, we can factor out an x from both terms: $x(x - 2) = 0$
  4. Solve for x: For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve:
    • x=0x = 0
    • x−2=0  ⟹  x=2x - 2 = 0 \implies x = 2

Therefore, the solutions for the equation x=0.5x2x = 0.5x^2 are x=0x = 0 and x=2x = 2. Looking at the provided options, it seems there might be a typo in the original question's answer choices, as none of them match the correct answer. The closest, and the intended correct option, would be if the options contained x=0x = 0 or x=2x = 2.

Explaining the Process

In this case, factoring was the most straightforward method. By rearranging the equation to the standard form and then factoring out the common factor x, we easily identified the two solutions. The key here is to bring all terms to one side, which allows us to set the equation equal to zero and apply the zero-product property (if a b = 0, then a = 0 or b = 0).

Solving the Quadratic Equation: 0=5x2−2x+60 = 5x^2 - 2x + 6

Now, let's move on to the second equation: 0=5x2−2x+60 = 5x^2 - 2x + 6. This one might not factor nicely, so we'll likely need to use either completing the square or the quadratic formula. Given the coefficients, the quadratic formula is a good choice.

Step-by-Step Solution

  1. Identify Coefficients: First, we identify the coefficients: a = 5, b = -2, and c = 6.
  2. Apply the Quadratic Formula: Plug these values into the quadratic formula: $x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $x = rac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(6)}}{2(5)}$
  3. Simplify: Simplify the expression: $x = rac{2 \pm \sqrt{4 - 120}}{10}$
  4. Simplify Further: $x = rac{2 \pm \sqrt{-116}}{10}$
  5. Deal with the Square Root of a Negative Number: The square root of a negative number means we'll have complex (imaginary) solutions. We can rewrite the square root: $x = rac{2 \pm \sqrt{116}i}{10}$
  6. Simplify the Square Root: Simplify 116{\sqrt{116}} to 229{2\sqrt{29}}. $x = rac{2 \pm 2\sqrt{29}i}{10}$
  7. Reduce the Fraction: Divide both the real and imaginary parts by 2: $x = rac{1 \pm \sqrt{29}i}{5}$

Result and Explanation

So, the solutions for the equation 0=5x2−2x+60 = 5x^2 - 2x + 6 are complex numbers: x = rac{1 + \sqrt{29}i}{5} and x = rac{1 - \sqrt{29}i}{5}. Looking at the provided options, it seems like there might be a typo in the original question's answer choices, as none of them match the correct answer. The closest, and the intended correct option, would be if the options contained these complex roots.

Why the Quadratic Formula?

The quadratic formula is a powerhouse here because factoring this equation is not straightforward. The coefficients don't lend themselves to easy factorization. Completing the square could also work, but the quadratic formula provides a direct and reliable route to the solution, especially when dealing with complex numbers. The presence of the negative value inside the square root indicates that the equation has no real roots, and we end up with complex roots.

Conclusion

Solving quadratic equations is a fundamental skill in mathematics, with applications that extend far beyond the classroom. We've explored two examples, demonstrating how to approach them using different methods. Remember to rearrange the equation to standard form, identify the coefficients, and choose the most appropriate method for solving. Practice regularly, and you'll find yourself becoming more confident in your ability to solve these equations. Keep in mind that understanding the principles behind each method is key, which will help you tackle a wide variety of problems! Happy solving, guys!