Solving M² + 6m = -4 With The Quadratic Formula Step-by-Step

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Hey guys! Let's dive into solving the quadratic equation m² + 6m = -4 using the quadratic formula. This is a classic problem in algebra, and mastering it will definitely boost your math skills. We'll break it down step-by-step, so don't worry if it seems intimidating at first. By the end of this article, you'll not only know how to solve this specific equation but also understand the general method for tackling any quadratic equation using the quadratic formula. So, grab your calculators and let's get started!

Understanding Quadratic Equations and the Quadratic Formula

Before we jump into solving our specific equation, let's quickly recap what quadratic equations are and what the quadratic formula is all about. Quadratic equations are polynomial equations of the second degree. This means the highest power of the variable (in our case, m) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. These equations pop up all over the place in math and science, from calculating projectile trajectories to modeling curves and shapes.

The quadratic formula is a super handy tool for finding the solutions (also called roots) of any quadratic equation. It provides a direct way to calculate the values of the variable that make the equation true. The formula itself looks like this:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • x represents the solutions (or roots) of the quadratic equation.
  • a, b, and c are the coefficients from the quadratic equation in the form ax² + bx + c = 0.
  • The ± symbol means there are actually two solutions: one where you add the square root part, and one where you subtract it. This is because quadratic equations can have up to two real solutions.

The part inside the square root, b² - 4ac, is called the discriminant. It tells us a lot about the nature of the solutions:

  • If b² - 4ac > 0, the equation has two distinct real solutions.
  • If b² - 4ac = 0, the equation has exactly one real solution (a repeated root).
  • If b² - 4ac < 0, the equation has two complex solutions (which involve imaginary numbers).

Knowing the quadratic formula and the discriminant is like having a superpower for solving quadratic equations. So, with this knowledge in our toolkit, let’s apply it to our specific problem and see how it works in action.

Step-by-Step Solution for m² + 6m = -4

Okay, let's tackle the equation m² + 6m = -4 head-on! The first thing we need to do when using the quadratic formula is to make sure our equation is in the standard form: ax² + bx + c = 0. Right now, we have m² + 6m = -4, so we need to move that -4 over to the left side. We can do this by simply adding 4 to both sides of the equation. This gives us:

m² + 6m + 4 = 0

Awesome! Now our equation is in the standard form, and we can clearly identify our coefficients:

  • a = 1 (the coefficient of m²)
  • b = 6 (the coefficient of m)
  • c = 4 (the constant term)

With a, b, and c in hand, we're ready to plug these values into the quadratic formula:

m = (-b ± √(b² - 4ac)) / 2a

Substituting our values, we get:

m = (-6 ± √(6² - 4 * 1 * 4)) / (2 * 1)

Now, let's simplify this step-by-step. First, we'll simplify the expression inside the square root:

  • 6² = 36
  • 4 * 1 * 4 = 16
  • 36 - 16 = 20

So our equation now looks like this:

m = (-6 ± √20) / 2

Next, we need to simplify the square root. We can simplify √20 by factoring out the largest perfect square. Since 20 = 4 * 5, and 4 is a perfect square (2² = 4), we can rewrite √20 as:

√20 = √(4 * 5) = √4 * √5 = 2√5

Substituting this back into our equation, we get:

m = (-6 ± 2√5) / 2

Finally, we can simplify the whole expression by dividing both terms in the numerator by 2:

m = -6/2 ± (2√5)/2

m = -3 ± √5

And there you have it! We've solved the equation. The solutions are m = -3 + √5 and m = -3 - √5. These are the two values of m that will make the original equation true. We did it, guys!

Identifying the Correct Answer and Why

Alright, now that we've walked through the solution, let's pinpoint the correct answer from the options provided. Remember, we found that the solutions to the equation m² + 6m = -4 are m = -3 + √5 and m = -3 - √5. This can be written in a more compact form as m = -3 ± √5.

Looking back at the options, we have:

a) m = -3 ± √5 b) m = 3 ± √5 c) m = -5 ± √3 d) m = 5 ± √3

It's clear that option a) m = -3 ± √5 matches our solution perfectly. So, that's the correct answer!

Now, let's briefly discuss why the other options are incorrect. This helps solidify our understanding and prevents us from making similar mistakes in the future.

  • Option b) m = 3 ± √5: This is incorrect because the sign of the -3 is wrong. Remember, in the quadratic formula, we have -b/2a, and in our case, b is 6, so -b/2a is -6/2 = -3, not 3.
  • Option c) m = -5 ± √3: This is incorrect because both the constant term and the value inside the square root are different. This likely results from errors in substituting the values into the quadratic formula or in simplifying the expression.
  • Option d) m = 5 ± √3: Similar to option c, this is incorrect because both the constant term and the value inside the square root are wrong. This indicates a misunderstanding of how to apply the quadratic formula or a mistake in the arithmetic.

By understanding why the correct answer is correct and why the incorrect answers are incorrect, we're not just memorizing a solution; we're building a deeper understanding of the process. This will make us much more confident and capable problem-solvers in the long run.

Real-World Applications of Quadratic Equations

So, we've nailed solving this quadratic equation, but you might be wondering,