Solving $x^2 + 8x - 1 = 0$ With The Quadratic Formula

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Hey guys! Today, we're diving into a classic algebra problem: solving a quadratic equation using the quadratic formula. We'll be tackling the equation $x^2 + 8x - 1 = 0$. Don't worry, it's not as scary as it looks! We'll break it down step-by-step and make sure you understand how to simplify your answer, including those pesky radicals.

Understanding the Quadratic Formula

Before we jump into the problem, let's quickly recap the quadratic formula. This formula is your best friend when you need to solve a quadratic equation in the standard form, which is $ax^2 + bx + c = 0$. The quadratic formula is:

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

Where:

• a is the coefficient of the $x^2$ term
• b is the coefficient of the $x$ term
• c is the constant term

The $\pm$ symbol means we'll get two possible solutions, one with a plus sign and one with a minus sign. This is because quadratic equations can have up to two real solutions.

Identifying a, b, and c

The first step in using the quadratic formula is correctly identifying the values of a, b, and c from our equation. In the equation $x^2 + 8x - 1 = 0$, we have:

• $a = 1$ (since the coefficient of $x^2$ is 1)
• $b = 8$ (the coefficient of $x$)
• $c = -1$ (the constant term)

Make sure you pay close attention to the signs! A negative sign can make a big difference in your final answer. Now that we've got a, b, and c, we're ready to plug them into the quadratic formula.

Applying the Quadratic Formula

Okay, let's substitute the values of a, b, and c into the quadratic formula:

$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-1)}}{2(1)}$

Now, we need to simplify this expression step-by-step. First, let's focus on the part under the square root, which is called the discriminant:

$8^2 - 4(1)(-1) = 64 + 4 = 68$

So, our equation now looks like this:

$x = \frac{-8 \pm \sqrt{68}}{2}$

Simplifying the Radical

Now, let's simplify the radical, $\sqrt{68}$. We need to find the largest perfect square that divides 68. The factors of 68 are 1, 2, 4, 17, 34, and 68. We see that 4 is the largest perfect square. So, we can rewrite $\sqrt{68}$ as:

$\sqrt{68} = \sqrt{4 \cdot 17} = \sqrt{4} \cdot \sqrt{17} = 2\sqrt{17}$

Now, substitute this back into our equation:

$x = \frac{-8 \pm 2\sqrt{17}}{2}$

Final Simplification

We're almost there! Notice that all the terms in the numerator and the denominator have a common factor of 2. We can divide each term by 2 to simplify:

$x = \frac{-8}{2} \pm \frac{2\sqrt{17}}{2} = -4 \pm \sqrt{17}$

So, our final solutions are:

$x = -4 + \sqrt{17}$ and $x = -4 - \sqrt{17}$

These are the two solutions to the quadratic equation $x^2 + 8x - 1 = 0$. We've successfully used the quadratic formula and simplified the answer, including the radical.

Breaking Down the Quadratic Formula: A More Detailed Look

Let's delve deeper into why the quadratic formula works and how each part contributes to finding the solutions. This understanding can help you remember the formula and apply it more confidently.

The Discriminant: The Key to Understanding Solutions

The discriminant, which is the part under the square root ($b^2 - 4ac$), is a crucial element of the quadratic formula. It tells us a lot about the nature of the solutions we'll find.

• If $b^2 - 4ac > 0$, the equation has two distinct real solutions (like in our example).
• If $b^2 - 4ac = 0$, the equation has one real solution (a repeated root).
• If $b^2 - 4ac < 0$, the equation has two complex solutions (involving imaginary numbers).

In our case, the discriminant was 68, which is greater than 0, confirming that we have two distinct real solutions. The discriminant essentially determines whether the parabola represented by the quadratic equation intersects the x-axis at two points, one point, or not at all.

The Importance of a, b, and c

The coefficients a, b, and c play significant roles in shaping the solutions of the quadratic equation. The coefficient a determines the parabola's direction (opens upwards if a > 0, downwards if a < 0) and its width. The coefficient b influences the axis of symmetry of the parabola, and c represents the y-intercept.

By correctly identifying these coefficients and plugging them into the quadratic formula, we're essentially using a pre-derived formula that encapsulates the process of completing the square for a general quadratic equation. The quadratic formula is derived by completing the square on the standard form equation $ax^2 + bx + c = 0$, which is why it works for any quadratic equation.

Why the $\pm$ Sign?

The $\pm$ sign is a direct consequence of taking the square root. When we solve for x after completing the square (the process that leads to the quadratic formula), we encounter a square root. Remember that a positive number has two square roots: a positive one and a negative one. This is why we have two possible solutions for x in a quadratic equation.

In our example, the $\pm \sqrt{17}$ part gives us two solutions: $-4 + \sqrt{17}$ and $-4 - \sqrt{17}$. These represent the two x-intercepts of the parabola $y = x^2 + 8x - 1$.

Common Mistakes to Avoid

Using the quadratic formula is pretty straightforward once you get the hang of it, but there are a few common pitfalls to watch out for:

1. Incorrectly identifying a, b, and c: Make sure you pay close attention to the signs and the coefficients. A wrong sign can throw off your entire solution.
2. Forgetting the $\pm$ sign: This is crucial for finding both solutions. Always remember to consider both the positive and negative square roots.
3. Simplifying the radical incorrectly: Make sure you find the largest perfect square factor before simplifying the radical. This will give you the simplest form of the solution.
4. Making arithmetic errors: Double-check your calculations, especially when dealing with negative numbers and fractions.
5. Not simplifying the final answer: Always simplify your answer as much as possible. This includes simplifying the radical and reducing any common factors.

Practice Makes Perfect

The best way to master the quadratic formula is to practice! Try solving different quadratic equations with varying coefficients and constants. Pay attention to the discriminant and how it affects the nature of the solutions. The more you practice, the more comfortable and confident you'll become with this powerful tool.

Example Problems for Practice

Here are a few more equations you can try solving using the quadratic formula:

1. $2x^2 - 5x + 2 = 0$
2. $x^2 + 6x + 9 = 0$
3. $3x^2 + 2x + 1 = 0$

Remember to identify a, b, and c, plug them into the formula, simplify the radical, and simplify your final answer.

Conclusion

So, there you have it! We've successfully solved the equation $x^2 + 8x - 1 = 0$ using the quadratic formula. We've also explored the formula in detail, discussed common mistakes to avoid, and provided some practice problems. The quadratic formula is a fundamental tool in algebra, and mastering it will open doors to solving a wide range of mathematical problems.

Keep practicing, and you'll be solving quadratic equations like a pro in no time! Good luck, guys, and happy problem-solving!