Finding The Zeros: Solving The Quadratic Equation

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Finding the Zeros of a Quadratic Function: A Step-by-Step Guide

Hey guys! Ever been stumped by a quadratic equation? Don't sweat it! Finding the zeros, also known as the roots or x-intercepts, is a fundamental skill in algebra. Today, we're diving deep into the quadratic function f(x) = 6x² + 12x - 7 and figuring out its zeros. We'll break it down so even if you're new to this, you'll feel like a math whiz by the end. Let's get started and demystify this problem together! Understanding how to find these zeros is super important for a bunch of things, from graphing to solving real-world problems. We're not just crunching numbers; we're building a solid foundation in math. So, grab your pencils, and let's go!

Understanding Quadratic Equations and Their Zeros

Alright, before we jump into the nitty-gritty, let's make sure we're all on the same page. A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic equation is a parabola, a U-shaped curve. The zeros of a quadratic function are the x-values where the function equals zero. In simpler terms, they are the x-intercepts, the points where the parabola crosses the x-axis. These are super important points because they tell us where the function's value changes from positive to negative or vice versa. They give us crucial insights into the behavior of the quadratic function. The zeros can be real numbers (meaning they can be plotted on a number line) or complex numbers (involving the imaginary unit i). When solving for zeros, we're essentially looking for the x-values that satisfy the equation. There are a few ways to find the zeros of a quadratic function: factoring, completing the square, and using the quadratic formula. Each method has its pros and cons, and which one you use often depends on the specific equation you're dealing with. Knowing all three will give you the edge.

Why Zeros Matter

  • Graphing: Zeros help you sketch the graph of the parabola accurately. They tell you where the curve crosses the x-axis. Knowing the zeros helps to determine the axis of symmetry, the vertex, and the direction the parabola opens (upwards or downwards).
  • Solving Equations: Zeros are solutions to the quadratic equation. They provide the values of x that make the equation true.
  • Real-World Applications: Quadratic equations model many real-world scenarios, such as the trajectory of a ball, the shape of a bridge, or the profit of a business. Zeros in these applications can represent critical points like when a ball hits the ground or when a business breaks even.

Solving the Quadratic Equation: f(x) = 6x² + 12x - 7

Now, let's get down to business and solve the quadratic equation f(x) = 6x² + 12x - 7 = 0. We'll be using the quadratic formula because it's the most reliable method for any quadratic equation, regardless of how complicated it looks. The quadratic formula is a lifesaver, and it's something you'll want to memorize. Here's what it looks like: x = (-b ± √(b² - 4ac)) / 2a. In our equation, a = 6, b = 12, and c = -7. Let's plug these values into the formula and see what happens.

Step-by-Step Solution

  1. Identify a, b, and c: In the equation 6x² + 12x - 7 = 0, we have a = 6, b = 12, and c = -7. Always make sure your equation is in the standard form first! If it's not, you'll need to rearrange the terms to match. Double-check your signs, too; a tiny mistake can lead to the wrong answer.
  2. Plug into the Quadratic Formula: Substitute the values into the quadratic formula: x = (-12 ± √(12² - 4 * 6 * -7)) / (2 * 6)
  3. Simplify the Expression: Now, we simplify: x = (-12 ± √(144 + 168)) / 12 x = (-12 ± √(312)) / 12
  4. Simplify the Square Root: Let's simplify the square root of 312. You can break it down into its prime factors. 312 = 2 * 2 * 2 * 3 * 13. Since we're looking for pairs, we have a pair of 2s, so we can pull a 2 out of the square root. √312 = √(4 * 78) = 2√78 So, now our formula looks like: x = (-12 ± 2√78) / 12
  5. Simplify the Fraction: We can simplify the fraction by dividing each term by 2: x = (-6 ± √78) / 6
  6. Further Simplification: Further divide each term by 6 x = -1 ± (√78)/6

Finding the Roots

Now we have our two solutions (zeros): x = -1 - √(78)/6 and x = -1 + √(78)/6. Comparing our solution with the options, let's go a step further and simplify the expression, by dividing by 6. So let's compare with the given options to find the correct answer. The options are in the simplified radical form. So the first step is to simplify the radical.

Simplify the radical term

x = (-12 ± √(312)) / 12 We already know that √312 = 2√78. So we need to simplify further. x = (-12 ± 2√78) / 12 Divide each term by the common factor, which is 2: x = (-6 ± √78) / 6 Therefore, x = -1 ± √78/6. The roots obtained from the quadratic formula do not match the options given. The best option is to review the solution, step by step, to find any mistakes.

Revisiting the Solution: Error Detection

It seems we've made an error in the previous calculation. Going back to where we plugged our values into the quadratic formula. Let's recalculate the root again.

x = (-12 ± √(12² - 4 * 6 * -7)) / (2 * 6) x = (-12 ± √(144 + 168)) / 12 x = (-12 ± √(312)) / 12

Now, let's simplify the square root of 312 correctly. 312 can be factored as 4 * 78, and further as 4 * 6 * 13. Thus, √312 simplifies to 2√78.

x = (-12 ± √(4 * 78)) / 12 x = (-12 ± 2√78) / 12 Simplify the fraction by dividing each term by 2: x = (-6 ± √78) / 6 x = -1 ± √78/6

Now it is still obvious that the result is not aligned with the given options, implying that there might have been a calculation error in the original given options. Since we cannot modify the given options, we can choose the best option available. Given the nature of square roots, the option A, B, and C can be expressed to a square root format. Based on our calculation, it is not possible to match the format of the options, with the given calculation. Let us consider the closest one, which is option A. Let us try to obtain the root by expressing it using the option A format.

Try expressing the root with the options

  • Option A: x = -1 ± √(13/6) To match this format, let us work on the final step. x = (-12 ± √(312)) / 12 x = (-12 ± √(144 + 168)) / 12 If we divide each term by 12, x = (-1 ± √(312/144)) x = (-1 ± √(13/6)) If you compare the final simplified expression with Option A: x = -1 ± √(13/6), we'll see that this does match perfectly.

So, the answer is A. This demonstrates how important it is to be thorough and double-check your work!

Conclusion: Mastering Quadratic Zeros

Alright, guys, you've done it! You've successfully found the zeros of the quadratic function f(x) = 6x² + 12x - 7. We've navigated through the quadratic formula and simplified our results. Remember, the zeros are where the parabola crosses the x-axis and are a key part of understanding quadratic functions. Keep practicing, and you'll become a pro at these problems in no time. If you're struggling, don't be afraid to go back over the steps and work through some more examples. The more you practice, the easier it becomes. You've got this! And always remember that math can be fun! Keep exploring, and you'll find it gets easier and more exciting with every step.