Solving Quadratic Equations Using The Quadratic Formula For 5x² + 3x - 4 = 0

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Hey guys! Ever stared at a quadratic equation and felt like you were trying to decipher ancient hieroglyphics? You're definitely not alone! Quadratic equations can seem intimidating, but once you understand the quadratic formula, they become much less scary. Today, we're diving deep into one such equation: 5x² + 3x - 4 = 0. Our mission? To correctly apply the quadratic formula and find the solutions for x. So, buckle up, and let's get started!

What is the Quadratic Formula?

Before we jump into solving our specific equation, let's quickly recap what the quadratic formula actually is. The quadratic formula is your trusty tool for solving any quadratic equation in the standard form of ax² + bx + c = 0. It's like a universal key that unlocks the solutions (also known as roots or zeros) for x. The formula itself looks like this:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • a is the coefficient of the x² term
  • b is the coefficient of the x term
  • c is the constant term

Think of it as a recipe – you plug in the right ingredients (a, b, and c), follow the steps (the formula), and voila! You get your solution(s) for x. The plus-minus (±) symbol is crucial because it tells us that there are typically two possible solutions to a quadratic equation. This is because the square root can result in both a positive and a negative value.

Breaking Down Our Equation: 5x² + 3x - 4 = 0

Now that we've refreshed our understanding of the quadratic formula, let's focus on our specific equation: 5x² + 3x - 4 = 0. The first step is to identify the values of a, b, and c. This is super straightforward:

  • a = 5 (the coefficient of x²)
  • b = 3 (the coefficient of x)
  • c = -4 (the constant term)

See? No sweat! We've got our ingredients ready. Now, it's time to plug them into our quadratic formula recipe. This is where careful attention to detail is key. A small mistake in substitution can throw off the entire calculation, leading to the wrong answer. So, let's take our time and do it right.

Plugging the Values into the Quadratic Formula

Okay, guys, here comes the main event: substituting our values into the quadratic formula. Remember the formula:

x = (-b ± √(b² - 4ac)) / 2a

Let's replace a, b, and c with their respective values:

x = (-3 ± √(3² - 4 * 5 * -4)) / (2 * 5)

Notice how we've carefully placed each value in its correct spot. The negative sign in front of the b is especially important, so don't forget about it! Also, pay close attention to the signs of the numbers. c is -4, so we need to include that negative sign in our calculation.

This is the point where many students might make a mistake, so double-checking your substitution is always a good idea. Make sure you've replaced each variable with the correct value and that you haven't missed any negative signs. Once you're confident in your substitution, it's time to simplify the expression.

Simplifying the Expression Step-by-Step

Now that we have our values plugged into the formula, it's time to simplify. This involves carefully following the order of operations (PEMDAS/BODMAS) to ensure we get the correct result. Let's break it down step by step:

  1. Simplify inside the square root:
    • First, calculate the square: 3² = 9
    • Then, multiply: 4 * 5 * -4 = -80
    • Now, subtract: 9 - (-80) = 9 + 80 = 89
    • So, our expression under the square root becomes √89
  2. Simplify the denominator:
    • 2 * 5 = 10
  3. Rewrite the entire expression:
    • x = (-3 ± √89) / 10

We've successfully simplified the expression! This form clearly shows us the two possible solutions for x. One solution involves adding the square root of 89 to -3, and the other involves subtracting the square root of 89 from -3, both divided by 10. This is the simplified form that directly answers the question of how the quadratic formula is correctly applied.

Identifying the Correct Option

Now, let's look at the answer choices provided in the question. We're looking for the option that matches our simplified expression:

x = (-3 ± √89) / 10

Comparing this to the options, we can clearly see that option A matches our result:

A. x = (-3 ± √(3² - 4(5)(-4))) / 2(5)

Option A correctly substitutes the values of a, b, and c into the quadratic formula. The other options either have incorrect signs or incorrect substitutions.

Why Other Options Are Incorrect

To solidify our understanding, let's briefly examine why the other options are incorrect:

  • Option B: x = (3 ± √(3² + 4(5)(-4))) / 2(5)
    • This option has the incorrect sign for the b term. It should be -3, not 3.
  • Option C: x = (3 ± √(3² - 4(5)(-4))) / 2(5)
    • This option also has the incorrect sign for the b term. It should be -3, not 3.
  • Option D: x = (-3 ± √(3² - 4(5)(-4))) / 2(5)
    • While this option correctly substitutes the values into the quadratic formula, it doesn't simplify the expression under the square root. This makes it harder to compare directly.

By understanding why these options are wrong, we reinforce our understanding of the correct application of the quadratic formula.

Key Takeaways and Common Mistakes to Avoid

Alright, guys, we've covered a lot! Let's recap the key takeaways and highlight some common mistakes to avoid when using the quadratic formula:

  • Key Takeaways:
    • The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a
    • Identify a, b, and c correctly from the equation ax² + bx + c = 0
    • Substitute the values carefully into the formula
    • Simplify the expression step-by-step, following the order of operations
    • Pay close attention to signs, especially the negative sign in front of b and the sign of c
  • Common Mistakes to Avoid:
    • Incorrectly identifying a, b, or c
    • Forgetting the negative sign in front of b
    • Making errors in arithmetic when simplifying the expression under the square root
    • Not following the order of operations
    • Confusing the ± symbol and only finding one solution

By keeping these points in mind, you'll be well-equipped to tackle any quadratic equation that comes your way!

Practice Makes Perfect: Quadratic Formula Exercises

The best way to master the quadratic formula is through practice! Here are a few extra exercises you can try:

  1. 2x² - 5x + 2 = 0
  2. x² + 4x - 7 = 0
  3. 3x² - 2x - 1 = 0

Work through these equations, carefully applying the quadratic formula. Check your answers using an online calculator or by graphing the quadratic function. Remember, the more you practice, the more confident you'll become!

Conclusion: Mastering the Quadratic Formula

So, there you have it! We've successfully navigated the quadratic formula and correctly solved the equation 5x² + 3x - 4 = 0. By understanding the formula, identifying the coefficients, substituting carefully, and simplifying step-by-step, you can conquer any quadratic equation. Remember to avoid common mistakes and practice regularly to build your skills. Keep up the great work, and you'll be a quadratic equation pro in no time! Guys, you've got this! Now go forth and solve those equations!