Solving Quadratic Equations: Step-by-Step Guide

Hey everyone! Today, we're diving into the world of quadratic equations. We'll be tackling several problems from the (6.10-6.17) set, providing you with detailed, step-by-step solutions and explanations. Whether you're a student struggling with algebra or just brushing up on your math skills, this guide is for you. Let's break down each equation, making sure you understand the 'how' and 'why' behind every step. Let's get started, guys!

Equation 6.10: Unveiling the Roots

We will now find the roots of the equations (6.10-6.17) and go through each problem one by one. Understanding the different methods to solve these equations is key. We'll begin by rewriting them in standard form and then applying the appropriate techniques. This section is specifically designed to help you, so you can easily understand and solve similar problems. We will make sure that the solutions are clear, concise, and easy to follow. Each step is explained in detail to ensure you grasp the concepts, making this a useful guide for students. Let's start with the first set of equations in 6.10!

1) 4x² - 3x = -3(12 - x)

First, we want to rewrite the equation in standard quadratic form (ax² + bx + c = 0). Let's start by expanding the right side and moving all terms to the left side: 4x² - 3x = -36 + 3x. Now, subtract 3x from both sides: 4x² - 6x + 36 = 0. We can simplify this a bit further by dividing the entire equation by 2: 2x² - 3x + 18 = 0. Now we can apply the quadratic formula. The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a, where a = 2, b = -3, and c = 18. Plugging in these values, we get x = (3 ± √((-3)² - 4 * 2 * 18)) / (2 * 2). This simplifies to x = (3 ± √(9 - 144)) / 4, or x = (3 ± √(-135)) / 4. Since the discriminant (b² - 4ac) is negative, the roots are complex. So the roots are x = (3 + i√135) / 4 and x = (3 - i√135) / 4. It's essential to understand how to manipulate the equation to achieve the form needed to apply the quadratic formula. Remember to always check your work!

2) 12x² - 5x = 3(12 - 2x) + x

Let’s start with the second equation. Expanding the right side gives us 12x² - 5x = 36 - 6x + x. Combining terms and rewriting in standard form: 12x² - 5x + 6x - x - 36 = 0. Simplifying this, we get 12x² - 0x - 36 = 0, which is equal to 12x² - 36 = 0. Divide both sides by 12, then we have x² - 3 = 0. The discriminant here is 0² - 4 * 1 * (-3) = 12. Use the formula: x = (0 ± √12) / 2. Which leads to x = ±√(12)/2 = ±2. So the solutions are x = √3, and x = -√3. Again, simplifying the original equation to find the roots, by using the simplest forms of the quadratic formula. Note that the simplification step is crucial to avoid calculation errors.

3) x² + 3x = 2(12 - x) - x

Expanding the right side, we get x² + 3x = 24 - 2x - x. Move all terms to the left side: x² + 3x + 2x + x - 24 = 0. Simplify: x² + 6x - 24 = 0. Applying the quadratic formula: x = (-6 ± √(6² - 4 * 1 * -24)) / 2. That simplifies to x = (-6 ± √(36 + 96)) / 2. Which in turn simplifies to x = (-6 ± √132) / 2. We can simplify √132 as 2√33. Thus, x = (-6 ± 2√33) / 2. So the roots are x = -3 + √33 and x = -3 - √33. The key here is not only to solve, but also to understand the steps involved in manipulating the equations.

4) 9x² - 3x = 3(12 - x)

Start by expanding: 9x² - 3x = 36 - 3x. Move all terms to the left: 9x² - 3x + 3x - 36 = 0. This simplifies to 9x² - 36 = 0. Divide both sides by 9: x² - 4 = 0. Then x² = 4, so x = ±2. The roots are x = 2 and x = -2. Using the quadratic formula is a direct approach, but simplification can often lead to a quicker solution. Always check if you can simplify the equation before applying the quadratic formula.

5) -x² + 6x = 2(12 + 2x) + 2x

Expanding the right side gives us -x² + 6x = 24 + 4x + 2x. Rearrange and simplify: -x² + 6x - 4x - 2x - 24 = 0. Which simplifies to -x² - 24 = 0. Multiplying both sides by -1: x² + 24 = 0. x² = -24. So, x = ±i√24 = ±2i√6. The solutions are complex numbers. Careful attention to detail is crucial when dealing with negative signs and simplifying.

6) 4x² - 3x = 36 - 3x

Let’s rearrange the equation: 4x² - 3x + 3x - 36 = 0. Simplifying to 4x² - 36 = 0. Dividing by 4: x² - 9 = 0. Thus, x² = 9, so x = ±3. The roots are x = 3 and x = -3. It’s important to practice these steps and to be comfortable in simplifying these equations to their simplest form. Always double-check your calculations to avoid common errors.

Key Takeaways and Tips

Throughout these problems, we've seen various techniques for solving quadratic equations. The most important tip is to practice. The more you work with these equations, the easier it will become. Here’s a summary of the techniques used:

• Standard Form: Always rewrite the equation in the form ax² + bx + c = 0.
• Quadratic Formula: Use it when factoring isn't straightforward. Remember the formula: x = (-b ± √(b² - 4ac)) / 2a.
• Simplification: Always look for opportunities to simplify the equation before applying the quadratic formula.
• Factoring: Try factoring if the equation is simple. This can be quicker than the quadratic formula.
• Check for Errors: After solving, always check your solutions by plugging them back into the original equation.

By following these steps, you'll be well on your way to mastering quadratic equations. Keep practicing, and don't be afraid to ask for help if you need it. This is a journey, and every step counts.

Conclusion: Your Path to Mastery

Guys, we've walked through the solutions to several quadratic equations. From the basics of rearranging terms to the application of the quadratic formula, you've seen how to tackle these problems step-by-step. Remember that the core of mastering these equations is consistency. Make sure to solve more problems on your own. Don't hesitate to revisit these examples whenever you need a refresher. Keep up the hard work, and you'll become confident in solving any quadratic equation that comes your way. So, keep practicing, keep learning, and keep growing! You've got this!