Solving Quadratic Equations Using The Quadratic Formula A Step-by-Step Guide

Jul 19, 2025 by ADMIN 77 views

Let's dive into solving quadratic equations using the quadratic formula, guys! It might seem a bit intimidating at first, but trust me, once you get the hang of it, it's super useful. We'll break it down step by step, making it easy to understand. So, grab your pencils, and let's get started!

Understanding the Quadratic Formula

The quadratic formula is your best friend when you're faced with an equation in the form of $ax^2 + bx + c = 0$ . This formula helps us find the values of $x$ that satisfy the equation, also known as the roots or solutions. Remember, the general form of a quadratic equation is crucial to identifying the coefficients $a$ , $b$ , and $c$ , which we'll plug into the formula. The formula itself looks like this:

$x = rac{-b \pm \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\begin\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\offers you a comprehensive and easy-to-understand guide on tackling them. You'll discover the formula, how to apply it, and work through a real-world example. We'll also cover common mistakes to avoid. By the end, you'll be solving quadratic equations like a pro!

The quadratic formula is expressed as:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where:

$a$ , $b$ , and $c$ are coefficients from the quadratic equation in the form $ax^2 + bx + c = 0$ .
The $\pm$ symbol indicates that there are two possible solutions, one using addition and one using subtraction.
The expression under the square root, $b^2 - 4ac$ , is known as the discriminant. This tells us about the nature of the roots (real, distinct, or complex). More on that later!

Key Takeaway: The quadratic formula provides a systematic method for solving any quadratic equation, regardless of whether it can be factored easily.

Applying the Quadratic Formula: A Step-by-Step Guide

Alright, let's get practical. Here's how you can use the quadratic formula to solve for $x$ in any quadratic equation. I'm breaking it down into simple steps to make it as clear as possible:

Identify a, b, and c: First things first, make sure your quadratic equation is in the standard form: $ax^2 + bx + c = 0$ . Then, carefully identify the coefficients $a$ , $b$ , and $c$ . Remember that $a$ is the coefficient of $x^2$ , $b$ is the coefficient of $x$ , and $c$ is the constant term. This is crucial; messing up the coefficients will lead to the wrong answer. So, double-check!
Plug the values into the formula: Now, substitute the values of $a$ , $b$ , and $c$ you've identified into the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This step is straightforward, but attention to detail is key. Be careful with the signs (positive and negative) of the coefficients. A common mistake is to forget the negative sign in front of $b$ in the formula.
Simplify: This is where you roll up your sleeves and do the math. Start by simplifying the expression under the square root (the discriminant): $b^2 - 4ac$ . This will give you a single number. Next, calculate the square root of the discriminant. If the discriminant is negative, you'll be dealing with imaginary numbers (more on that later). Finally, simplify the entire expression, following the order of operations (PEMDAS/BODMAS). This usually involves performing the multiplication and division before the addition and subtraction.
Calculate the two solutions: Remember that $\pm$ sign? It means you'll have two possible solutions for $x$ . Calculate them separately:
- $x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$
- $x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$
These are your two roots or solutions to the quadratic equation.
Check your solutions: It's always a good idea to check your answers. Plug each solution back into the original quadratic equation to make sure it satisfies the equation. If it does, you've found the correct solutions!

Key Takeaway: Applying the quadratic formula involves a systematic process of identifying coefficients, substituting them into the formula, simplifying, and calculating the solutions. Each step is important, and accuracy is key.

Example: Solving $x^2 - 11x + 28 = 0$

Let's put this into practice with the equation you've given: $x^2 - 11x + 28 = 0$ . We'll go through the steps together, and you'll see how easy it is. I promise!

Identify a, b, and c: In this equation:
- $a = 1$ (the coefficient of $x^2$ )
- $b = -11$ (the coefficient of $x$ )
- $c = 28$ (the constant term)
See? That wasn't so bad, right? Remember to pay close attention to the signs. The negative sign in front of 11 is crucial.
Plug the values into the formula: Now, let's substitute these values into the quadratic formula:

$x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(1)(28)}}{2(1)}$

Notice how we carefully substituted each value into its corresponding place in the formula. The double negative in front of 11 is important – it'll become positive in the next step.
Simplify: Time to simplify the expression. Let's start with the discriminant:

$(-11)^2 - 4(1)(28) = 121 - 112 = 9$

So, the discriminant is 9. Now, let's calculate the square root:

$\sqrt{9} = 3$

Great! Now we can plug this back into the formula:

$x = \frac{11 \pm 3}{2}$
Calculate the two solutions: Now, we'll calculate the two possible solutions for $x$ :
- $x_1 = \frac{11 + 3}{2} = \frac{14}{2} = 7$
- $x_2 = \frac{11 - 3}{2} = \frac{8}{2} = 4$
So, our two solutions are $x = 7$ and $x = 4$ .
Check your solutions: Let's make sure our answers are correct. We'll plug each solution back into the original equation:
- For $x = 7$ : $(7)^2 - 11(7) + 28 = 49 - 77 + 28 = 0$ (Correct!)
- For $x = 4$ : $(4)^2 - 11(4) + 28 = 16 - 44 + 28 = 0$ (Correct!)
Both solutions satisfy the equation, so we've done it correctly!

Key Takeaway: This example demonstrates how to apply the quadratic formula step-by-step. By carefully substituting values, simplifying, and calculating the solutions, you can confidently solve any quadratic equation.

Common Mistakes to Avoid

Alright, guys, let's talk about some common pitfalls when using the quadratic formula. Knowing these mistakes ahead of time can save you a lot of headaches. Trust me, I've seen these mistakes plenty of times!

Incorrectly identifying a, b, and c: This is the most common mistake, hands down. As we discussed earlier, it's crucial to correctly identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation. A simple sign error or misidentification can throw off your entire solution. Always double-check your values before plugging them into the formula.
Sign errors: Negative signs can be tricky little buggers. Be extra careful when dealing with negative coefficients or when substituting values into the formula. For example, forgetting the negative sign in front of the $b$ in the formula ( $-b$ ) is a classic mistake. Similarly, mishandling the negative sign under the square root (the discriminant) can lead to incorrect solutions or complex numbers when you shouldn't have them.
Order of operations: Remember PEMDAS/BODMAS! You need to simplify the expression under the square root first ( $b^2 - 4ac$ ), then calculate the square root, and then perform the other operations in the correct order. Mixing up the order of operations will lead to an incorrect result. Stick to the rules, and you'll be golden.
Forgetting the $\pm$ sign: The $\pm$ sign in the quadratic formula is there for a reason – it indicates that there are two possible solutions. Don't forget to calculate both solutions by using both the plus and minus signs. If you only calculate one solution, you're only getting half the answer.
Not simplifying completely: Make sure you simplify your solutions as much as possible. This might involve reducing fractions, simplifying radicals, or combining like terms. A solution that isn't fully simplified is like a job that's only half done. Take the extra step to get the most simplified answer.

Key Takeaway: Being aware of these common mistakes and taking steps to avoid them will significantly improve your accuracy and confidence in solving quadratic equations using the quadratic formula.

Wrapping Up

So, guys, we've covered a lot in this guide. You've learned what the quadratic formula is, how to apply it step by step, worked through a real-world example, and learned about common mistakes to avoid. Now, you're well-equipped to tackle any quadratic equation that comes your way. Remember, practice makes perfect. The more you use the quadratic formula, the more comfortable and confident you'll become.

The solutions to the equation $x^2 - 11x + 28 = 0$ are $x = 4$ and $x = 7$ . Since the question asks for the smallest solution first, the answer is $x = 4$ .

Keep practicing, and you'll be a quadratic equation-solving wizard in no time!