Solving Quadratics: The Best First Step

When you're faced with a quadratic equation like $x^2 + 2x + 13 = -8$, it can feel a bit daunting at first. But don't worry, breaking it down into manageable steps makes it much easier. So, what's the most logical first move here? Let's walk through the options and see which one sets us up for success.

Understanding the Options

Before diving into the best approach, let's consider each of the given options:

• A. Add 8 to both sides: This involves adding 8 to both sides of the equation, resulting in $x^2 + 2x + 21 = 0$.
• B. Take the square root of both sides: This option isn't directly applicable since we can't isolate a perfect square term easily at this stage.
• C. Divide both sides by $x$: Dividing by $x$ would change the fundamental nature of the equation and potentially eliminate a solution ($x = 0$), so it’s not a good idea.
• D. Subtract 13 from both sides: Subtracting 13 from both sides gives us $x^2 + 2x = -21$, which doesn't immediately simplify our problem.

The Importance of Setting Up the Equation Correctly

When tackling quadratic equations, the initial setup is crucial. The goal is to manipulate the equation into a standard form that allows us to apply well-known solution methods. The most common forms are:

1. Standard Form: $ax^2 + bx + c = 0$
2. Vertex Form: $a(x - h)^2 + k = 0$

The standard form is particularly useful because it allows us to use methods like factoring, completing the square, or the quadratic formula. The vertex form is great for identifying the vertex of the parabola represented by the quadratic equation. Knowing these forms helps you make informed decisions about your first steps.

Why Adding 8 Is the Logical First Step

The most logical first step is A. Add 8 to both sides. Here’s why:

Adding 8 to both sides of the equation $x^2 + 2x + 13 = -8$ transforms it into:

$x^2 + 2x + 13 + 8 = -8 + 8$

Which simplifies to:

$x^2 + 2x + 21 = 0$

This resulting equation is now in the standard form $ax^2 + bx + c = 0$, where $a = 1$, $b = 2$, and $c = 21$. By getting the equation into this form, we open the door to several solution methods:

• Quadratic Formula: The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, is a reliable method for finding the roots of any quadratic equation in standard form. In this case, it would be $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(21)}}{2(1)}$.
• Completing the Square: Completing the square involves manipulating the equation to form a perfect square trinomial. While it might not be the most straightforward method for this particular equation, it’s still a viable option.
• Factoring: Although not always possible, factoring is the quickest method when it works. In this case, we would look for two numbers that multiply to 21 and add to 2. However, since there are no such real numbers, factoring doesn't work in the traditional sense here.

Detailed Explanation

Let's dive deeper into why adding 8 is the right move. The original equation is $x^2 + 2x + 13 = -8$. Our primary goal is to set the equation to zero, which is a prerequisite for applying the quadratic formula or attempting to factor. When we add 8 to both sides, we effectively neutralize the -8 on the right side, bringing us closer to the standard form.

• Isolating Terms: By adding 8, we consolidate all the constant terms on one side, making it easier to analyze the equation's structure. This is a fundamental principle in solving any equation—group like terms together.
• Preparing for Solution Methods: The standard form $ax^2 + bx + c = 0$ is specifically designed to work with the quadratic formula. Adding 8 is the direct step toward achieving this form. Once in this form, you can directly plug the coefficients into the formula and solve for x.
• Avoiding Unnecessary Complexity: Other options, like dividing by x or subtracting 13, don't lead us closer to a solvable form. Dividing by x is especially problematic because it assumes $x \neq 0$ and can eliminate potential solutions. Subtracting 13 just moves the constant term around without simplifying the overall equation.

Practical Steps After Adding 8

Once you've added 8 to both sides and have the equation $x^2 + 2x + 21 = 0$, here’s how you can proceed:

1. Apply the Quadratic Formula:

   • Identify $a = 1$, $b = 2$, and $c = 21$.
   • Plug these values into the quadratic formula: $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(21)}}{2(1)}$ $x = \frac{-2 \pm \sqrt{4 - 84}}{2}$ $x = \frac{-2 \pm \sqrt{-80}}{2}$
   • Simplify the square root of the negative number using imaginary numbers: $x = \frac{-2 \pm 4i\sqrt{5}}{2}$
   • Further simplify to get the final solutions: $x = -1 \pm 2i\sqrt{5}$

2. Check for Factorability (Optional):

   • In this case, you'll find that there are no real factors for 21 that add up to 2, confirming that factoring isn't a straightforward method here.

3. Completing the Square (Alternative):

   • Rewrite the equation as $x^2 + 2x = -21$.
   • Add $(b/2)^2 = (2/2)^2 = 1$ to both sides: $x^2 + 2x + 1 = -21 + 1$.
   • Factor the left side: $(x + 1)^2 = -20$.
   • Take the square root of both sides: $x + 1 = \pm \sqrt{-20}$.
   • Solve for $x$: $x = -1 \pm 2i\sqrt{5}$.

Common Mistakes to Avoid

When solving quadratic equations, here are some common pitfalls to watch out for:

• Dividing by $x$: This is generally a bad idea unless you know for sure that $x \neq 0$. You risk losing one of the solutions.
• Incorrectly Applying the Quadratic Formula: Make sure you correctly identify $a$, $b$, and $c$, and plug them into the formula carefully.
• Forgetting the $\pm$ Sign: When taking the square root, always remember to consider both positive and negative roots.
• Making Arithmetic Errors: Simple arithmetic mistakes can throw off your entire solution, so double-check your work at each step.

Conclusion

In summary, for the quadratic equation $x^2 + 2x + 13 = -8$, the most logical first step is to add 8 to both sides. This brings the equation into the standard form $ax^2 + bx + c = 0$, which sets you up perfectly for using the quadratic formula, completing the square, or attempting to factor. By following this approach, you’ll be well on your way to finding the solutions to the equation. Remember, understanding the structure of the equation and choosing the right initial steps can greatly simplify the problem-solving process.

For further reading on quadratic equations and their solutions, check out resources like Khan Academy's Quadratic Equations Section.