Solving Quadratic Equations: Step-by-Step With The Quadratic Formula

by Admin 69 views
Solving Quadratic Equations: Step-by-Step with the Quadratic Formula

Hey math enthusiasts! Ever feel like quadratic equations are a bit of a puzzle? Fear not, because today, we're going to crack the code and learn how to solve them using the mighty quadratic formula. This is a super handy tool that lets you find the values of x in any quadratic equation, no matter how complex it looks. We'll break down the process step-by-step, making it easy to understand and apply. We will also get to the bottom of the question: Using the quadratic formula to solve 4x2−3x+9=2x+14 x^2-3 x+9=2 x+1, what are the values of xx?

Understanding Quadratic Equations

First off, what exactly is a quadratic equation? Well, it's an equation that can be written in the form: $ax^2 + bx + c = 0$. Here, a, b, and c are constants (numbers), and x is the variable we're trying to find. The key feature is the x squared term (x2x^2). That's what makes it a quadratic! Quadratic equations can have two solutions, one solution, or no real solutions (we'll see what that means later). They pop up everywhere in the real world. For example, the path of a ball thrown in the air follows a quadratic equation. The graph of a quadratic equation is a parabola, a U-shaped curve. This is the foundation upon which everything else is built. If you can master this concept, you can solve many problems.

Why the Quadratic Formula?

You might be wondering, why not just factor the equation? Well, factoring is a great method, but it doesn't always work. Some quadratic equations are just too tricky to factor easily. That's where the quadratic formula comes to the rescue! It's a universal solution, meaning it works for any quadratic equation, no matter how complicated. With this tool in your arsenal, you'll be able to solve practically any quadratic equation that comes your way. It is a powerful tool and makes the process of solving quadratic equations more efficient and reliable. There's no more trial and error, just a straightforward path to the solution. The quadratic formula is a time-saver and a problem-solver, making complex equations manageable.

The Quadratic Formula Unveiled

Alright, let's get down to the formula itself. Here it is in all its glory:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Don't let it intimidate you! It looks a little scary at first, but we'll break it down piece by piece. Essentially, the formula tells us that x is equal to the following: negative b, plus or minus the square root of b squared minus 4 times a times c, all divided by 2 times a. Notice the plus or minus symbol (±\pm). That means there are potentially two solutions for x: one where we add the square root and another where we subtract it. Remember the letters a, b, and c? They come from the standard form of the quadratic equation (ax2+bx+c=0ax^2 + bx + c = 0). To use the quadratic formula, you'll need to identify the values of a, b, and c in the equation you're trying to solve. Once you know these values, just plug them into the formula, and you are golden. The formula is a recipe; you have to follow the instructions and put in the correct ingredients.

Breaking Down the Formula

Let's understand each part of the quadratic formula:

  • -b: This is the opposite of the b value in your equation. If b is positive, you'll use a negative value, and vice versa.
  • ±\pm (plus or minus): This is the secret to finding two possible solutions for x. You'll perform the calculations twice, once adding the square root and once subtracting it.
  • b2−4ac\sqrt{b^2 - 4ac}: This is the square root part. The expression inside the square root (b2−4acb^2 - 4ac) is called the discriminant. It tells us about the nature of the solutions. If the discriminant is positive, you'll get two real solutions. If it's zero, you'll get one real solution. If it's negative, you'll get two complex (imaginary) solutions.
  • 2a: This is two times the a value from your equation. You divide the entire result by this value.

Understanding each part of the formula provides a solid understanding of the nature of the solutions.

Step-by-Step Guide to Solving Quadratic Equations

Now, let's walk through the steps to solve a quadratic equation using the quadratic formula. We will break down each step so that you have a solid understanding of the process.

Step 1: Standard Form

The first step is to ensure your quadratic equation is in standard form (ax2+bx+c=0ax^2 + bx + c = 0). If it's not, rearrange the terms to get it into this form. For example, if you have something like 2x2+5x=32x^2 + 5x = 3, you'll need to subtract 3 from both sides to get 2x2+5x−3=02x^2 + 5x - 3 = 0.

Step 2: Identify a, b, and c

Once your equation is in standard form, identify the values of a, b, and c. Remember that a is the coefficient of the x2x^2 term, b is the coefficient of the x term, and c is the constant term. In our example 2x2+5x−3=02x^2 + 5x - 3 = 0, a = 2, b = 5, and c = -3. Double-check your values to avoid errors.

Step 3: Plug into the Formula

Now, substitute the values of a, b, and c into the quadratic formula. Be careful with signs (especially negative signs) when plugging in the values. For our example, the formula becomes: x=−5±52−4⋅2⋅−32⋅2x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 2 \cdot -3}}{2 \cdot 2}. This step is critical; ensuring correct substitution sets the stage for accurate solutions.

Step 4: Simplify

Carefully simplify the expression. First, calculate the value inside the square root (the discriminant). Then, simplify the rest of the equation. Continue simplifying until you have the final values for x. In our example, we'd simplify to get: x=−5±25+244x = \frac{-5 \pm \sqrt{25 + 24}}{4}, which simplifies to x=−5±494x = \frac{-5 \pm \sqrt{49}}{4}, and then to x=−5±74x = \frac{-5 \pm 7}{4}.

Step 5: Calculate the Solutions

Finally, calculate the two possible solutions for x. One solution will be where you add the square root, and the other will be where you subtract it. For our example, x=−5+74=24=0.5x = \frac{-5 + 7}{4} = \frac{2}{4} = 0.5 and x=−5−74=−124=−3x = \frac{-5 - 7}{4} = \frac{-12}{4} = -3. So, the solutions for our example equation are x = 0.5 and x = -3. Remember to double-check your calculations to ensure accuracy.

Solving Your Specific Equation

Let's get back to your original question: Using the quadratic formula to solve 4x2−3x+9=2x+14x^2 - 3x + 9 = 2x + 1, what are the values of xx? We will find the solution step by step.

Step 1: Standard Form

First, we need to rewrite the equation in standard form. Subtract 2x2x and 11 from both sides to get: $4x^2 - 3x - 2x + 9 - 1 = 0$ which simplifies to: $4x^2 - 5x + 8 = 0$

Step 2: Identify a, b, and c

Now, let's identify a, b, and c from our standard form equation: $4x^2 - 5x + 8 = 0$

  • a = 4
  • b = -5
  • c = 8

Step 3: Plug into the Formula

Let's plug these values into the quadratic formula: $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 4 \cdot 8}}{2 \cdot 4}$

Step 4: Simplify

Now, let's simplify the equation: $x = \frac{5 \pm \sqrt{25 - 128}}{8}$ $x = \frac{5 \pm \sqrt{-103}}{8}$

Step 5: Calculate the Solutions

Since the discriminant is negative (-103), we'll have complex (imaginary) solutions. Remember that the square root of -1 is represented as i. So, our solutions are: $x = \frac{5 \pm \sqrt{103}i}{8}$

Therefore, the correct answer is C. 5±103i8\frac{5 \pm \sqrt{103} i}{8}

Additional Tips for Success

  • Double-Check Your Work: Always review your steps to avoid careless mistakes. It's easy to make a small error, especially with negative signs.
  • Practice Makes Perfect: The more you practice, the more comfortable you'll become with the quadratic formula. Try solving different quadratic equations on your own.
  • Understand the Discriminant: Knowing about the discriminant (b2−4acb^2 - 4ac) can tell you whether your solutions will be real, or complex, and how many solutions you should expect.
  • Use a Calculator: Don't hesitate to use a calculator to help with calculations, especially when dealing with square roots or large numbers. Make sure you know how to correctly use your calculator to find square roots and perform arithmetic operations.

Conclusion

Congratulations, you have now mastered the art of solving quadratic equations using the quadratic formula! It might seem like a lot to take in at first, but with practice, it will become second nature. Remember to stay organized, be patient, and don't be afraid to ask for help if you get stuck. Keep practicing, and you'll be solving complex quadratic equations like a pro in no time! So go out there and conquer those quadratic equations, guys!