Solving 5x + 3x² = 4x - 7 A Quadratic Equation Example

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Hey guys! Let's dive into solving a quadratic equation step-by-step. We'll tackle the equation 5x + 3x² = 4x - 7, transform it into the standard form, identify the coefficients, and then use the famous Bhaskara's formula to find the roots. Buckle up, it's gonna be a fun ride!

Transforming the Equation into Standard Form

The first step in solving any quadratic equation is to get it into the standard form, which is ax² + bx + c = 0. This form is super important because it allows us to easily identify the coefficients a, b, and c, which we need for Bhaskara's formula. So, let's take our equation, 5x + 3x² = 4x - 7, and rearrange it.

To get everything on one side, we'll subtract 4x from both sides and add 7 to both sides. This gives us:

3x² + 5x - 4x + 7 = 0

Now, let's simplify by combining the x terms:

3x² + x + 7 = 0

Alright! We've successfully transformed the equation into the standard form. Now we can clearly see the coefficients. Let's identify them.

Identifying the Coefficients: a, b, and c

In our standard form equation, 3x² + x + 7 = 0, the coefficients are:

  • a = 3 (the coefficient of the x² term)
  • b = 1 (the coefficient of the x term)
  • c = 7 (the constant term)

So, we've nailed the first part! We have our equation in the standard form, and we know the values of a, b, and c. This is crucial for using Bhaskara's formula, which is our next adventure.

Applying Bhaskara's Formula to Find the Roots

Now comes the exciting part – using Bhaskara's formula to find the roots of our quadratic equation. Bhaskara's formula is a powerful tool that helps us solve any quadratic equation. For those who need a quick refresher, the formula is:

x = (-b ± √(b² - 4ac)) / 2a

It might look a bit intimidating at first, but don't worry! We'll break it down step-by-step. We already know the values of a, b, and c:

  • a = 3
  • b = 1
  • c = 7

Let's plug these values into the formula.

Step-by-Step Calculation

  1. Substitute the values:

    x = (-1 ± √(1² - 4 * 3 * 7)) / (2 * 3)

  2. Simplify the expression under the square root (the discriminant):

    The discriminant (Δ) is b² - 4ac. So, let's calculate it:

    Δ = 1² - 4 * 3 * 7 = 1 - 84 = -83

    Uh oh! We've hit a snag. The discriminant is negative (-83). This means the roots of the equation are complex numbers. Don't panic! Complex numbers are just numbers that have a real part and an imaginary part. It just means our solutions will involve the imaginary unit i, where i² = -1.

  3. Continue with the formula, incorporating the imaginary unit:

    x = (-1 ± √(-83)) / 6

    Since √(-83) = √(83 * -1) = √(83) * √(-1) = √(83) * i, we can rewrite the equation as:

    x = (-1 ± √(83)i) / 6

  4. Express the roots in the form a + bi:

    We have two roots:

    • x₁ = (-1 + √(83)i) / 6 = -1/6 + (√(83)/6)i
    • x₂ = (-1 - √(83)i) / 6 = -1/6 - (√(83)/6)i

The Roots of the Equation

So, the roots of the equation 5x + 3x² = 4x - 7 are complex numbers:

  • x₁ = -1/6 + (√(83)/6)i
  • x₂ = -1/6 - (√(83)/6)i

These are our final answers! We've successfully navigated through transforming the equation, identifying coefficients, and using Bhaskara's formula to find the complex roots. Awesome job, guys!

Conclusion

In this exercise, we took a quadratic equation, transformed it into the standard form ax² + bx + c = 0, identified the coefficients (a = 3, b = 1, c = 7), and then used Bhaskara's formula to find the roots. We encountered a negative discriminant, which led us to complex roots. This just shows how versatile quadratic equations can be! Understanding these steps is crucial for solving a wide range of mathematical problems. Keep practicing, and you'll become a quadratic equation-solving pro in no time!

Remember, the key is to break down the problem into smaller, manageable steps. Don't be intimidated by the formulas; just take it one step at a time, and you'll get there. Keep up the great work!

Analyzing the Given Alternatives

Now that we've solved the equation, let's take a look at the alternatives provided:

A) a = 3, b = 1, c = 7 B) a = 3, b = -1, c = ... (The alternative is incomplete in the original question)

We found that the coefficients are indeed a = 3, b = 1, and c = 7. So, alternative A is correct in identifying the coefficients. However, the original question also asks for the roots, which we found to be complex numbers. Since the alternatives only focus on the coefficients, we've gone a step further by actually solving for the roots.

Importance of Quadratic Equations

Quadratic equations aren't just abstract mathematical concepts; they have real-world applications in various fields. From physics (projectile motion) to engineering (designing structures) to finance (modeling investments), quadratic equations pop up everywhere! Mastering the art of solving these equations opens doors to understanding and solving problems in many different domains.

For example, consider the trajectory of a ball thrown into the air. The path it follows can be modeled by a quadratic equation. By solving the equation, we can determine the maximum height the ball reaches or how far it travels before hitting the ground. Similarly, in engineering, quadratic equations are used to calculate the optimal shape of bridges and other structures to ensure stability and minimize stress.

The more you explore mathematics, the more you'll realize how interconnected different concepts are. Quadratic equations are a fundamental building block for more advanced topics, so investing time in understanding them is definitely worth it. Keep exploring, keep learning, and keep having fun with math!

Further Practice and Exploration

If you're feeling confident, why not try solving some more quadratic equations? You can find plenty of practice problems online or in textbooks. Try varying the coefficients and see how it affects the roots. You can also explore different methods for solving quadratic equations, such as completing the square or factoring. Each method has its own advantages and disadvantages, and understanding them all will give you a more complete picture.

Also, consider delving deeper into the concept of complex numbers. They might seem a bit strange at first, but they are a fascinating and important part of mathematics. Complex numbers have applications in areas like electrical engineering, quantum mechanics, and even computer graphics!

Remember, the key to mastering any mathematical concept is practice and exploration. Don't be afraid to make mistakes; they are a natural part of the learning process. The more you practice, the more comfortable and confident you'll become. So, keep challenging yourself and keep exploring the wonderful world of mathematics!