Solve Quadratic Equations Using The Formula: A Step-by-Step Guide

by Omar Yusuf 66 views

Hey guys! Today, we're going to dive deep into solving quadratic equations using the quadratic formula. It might sound intimidating, but trust me, it's a super handy tool to have in your math arsenal. We'll take a specific equation as an example and break down each step, so you’ll feel like a pro in no time!

Understanding the Quadratic Formula

First off, let’s talk about what a quadratic equation actually is. In simple terms, a quadratic equation is an equation that can be written in the general form of ax² + bx + c = 0, where a, b, and c are constants (numbers), and a is not equal to zero. The x represents the variable we’re trying to solve for. These equations pop up everywhere in math and real-world applications, from calculating the trajectory of a ball to designing efficient structures.

Now, the quadratic formula is the magical key we use to find the solutions (also called roots or zeros) of these equations. It’s given by:

x = (-b ± √(b² - 4ac)) / (2a)

This formula might look a bit scary at first glance, but we’ll break it down piece by piece. The ± symbol means we actually have two solutions: one where we add the square root part and one where we subtract it. The part under the square root, b² - 4ac, is called the discriminant, and it tells us a lot about the nature of the solutions. If the discriminant is positive, we have two real solutions; if it's zero, we have one real solution (a repeated root); and if it's negative, we have two complex solutions.

Example Equation: x(-2x - 11) = -3

Let's take our example equation: x(-2x - 11) = -3. Before we can use the quadratic formula, we need to get it into the standard form, ax² + bx + c = 0. To do this, we'll first distribute the x on the left side and then move the constant term to the left side as well.

Step 1: Distribute the x

Multiply x by both terms inside the parentheses:

x(-2x) + x(-11) = -3

This simplifies to:

-2x² - 11x = -3

Step 2: Move the constant term to the left side

To get the equation in the standard form, we need to add 3 to both sides:

-2x² - 11x + 3 = 0

Now we have our quadratic equation in the standard form: -2x² - 11x + 3 = 0. Great! We're one step closer to solving it.

Identifying a, b, and c

Okay, so now that we have our equation in the standard form ax² + bx + c = 0, we need to identify the values of a, b, and c. This is super important because these are the numbers we're going to plug into the quadratic formula. Let's take a look at our equation again:

-2x² - 11x + 3 = 0

  • The coefficient of the x² term is a, so in this case, a = -2. Remember to include the negative sign if there is one!
  • The coefficient of the x term is b, so here, b = -11. Again, don't forget the negative sign.
  • The constant term is c, so c = 3.

So, to recap, we've identified our coefficients as a = -2, b = -11, and c = 3. These are the key ingredients we need for the quadratic formula recipe!

Plugging Values into the Quadratic Formula

Alright, guys, this is where the magic happens! We've got our values for a, b, and c, and we've got the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Now it's time to plug those values in and see what we get.

Step 1: Substitute the values

Replace a, b, and c in the formula with their respective values:

x = (-(-11) ± √((-11)² - 4(-2)(3))) / (2(-2))

Notice how we've carefully substituted each value, making sure to keep track of those negative signs. This is a common place where mistakes can happen, so it's always good to double-check your work.

Step 2: Simplify the expression

Now, let's start simplifying. First, let's deal with the double negative in front of the 11:

x = (11 ± √((-11)² - 4(-2)(3))) / (2(-2))

Next, let's calculate the square of -11 and the product inside the square root:

x = (11 ± √(121 - (-24))) / (-4)

Remember that subtracting a negative is the same as adding a positive, so:

x = (11 ± √(121 + 24)) / (-4)

Now we can add the numbers inside the square root:

x = (11 ± √145) / (-4)

And finally, we multiply in the denominator:

x = (11 ± √145) / (-4)

So, there we have it! We've plugged our values into the quadratic formula and simplified as much as we can. Now, let's look at how we can express our final answer.

Expressing the Solution

We've arrived at the solution: x = (11 ± √145) / (-4). This is actually two solutions in one! Remember the ± symbol means we have two possibilities: one where we add the square root and one where we subtract it.

Solution 1: Adding the square root

x₁ = (11 + √145) / (-4)

Solution 2: Subtracting the square root

x₂ = (11 - √145) / (-4)

These are the exact solutions to our quadratic equation. If we need decimal approximations, we can use a calculator to find the square root of 145 and perform the calculations. However, it's often best to leave the answer in this form unless a decimal approximation is specifically requested.

Now, let's think about how the original question asked us to present the answer. It wanted us to replace m, n, and p in the expression x = (m ± √n) / p with the correct values.

Comparing this with our solution, x = (11 ± √145) / (-4), we can easily identify the values:

  • m = 11
  • n = 145
  • p = -4

So, there you have it! We've successfully used the quadratic formula to solve the equation and expressed our solution in the requested format.

Key Takeaways and Tips

Solving quadratic equations using the quadratic formula might seem like a lot of steps, but with practice, it becomes second nature. Here are some key takeaways and tips to keep in mind:

  • Standard Form is Key: Always make sure your equation is in the standard form ax² + bx + c = 0 before identifying a, b, and c.
  • Careful with Signs: Pay extra attention to negative signs when substituting values into the formula. This is a common source of errors.
  • Simplify Step-by-Step: Break the problem down into smaller, manageable steps. This makes the process less overwhelming and reduces the chance of mistakes.
  • The Discriminant: Remember that the discriminant (b² - 4ac) tells you about the nature of the solutions. A positive discriminant means two real solutions, a zero discriminant means one real solution, and a negative discriminant means two complex solutions.
  • Practice Makes Perfect: The more you practice, the more comfortable you'll become with using the quadratic formula. Try solving different equations with varying coefficients.

So, guys, that's it for today! I hope this breakdown of the quadratic formula has been helpful. Remember, math is like any other skill – the more you practice, the better you get. Keep practicing, and you'll be solving quadratic equations like a pro in no time!