Solving X^2+4x+8=0 By Completing The Square

Introduction to Solving Quadratic Equations

When you're faced with a quadratic equation, especially one that doesn't easily factor, techniques like completing the square become your best friends. This method allows us to transform a standard quadratic equation into a form where we can isolate the variable and find its solutions. Today, we're going to tackle the equation $x^2+4x+8=0$ using this powerful technique and uncover its solutions.

Quadratic equations are equations of the form $ax^2+bx+c=0$, where $a$, $b$, and $c$ are constants and $a eq 0$. They are fundamental in algebra and appear in various fields, from physics and engineering to economics and statistics. While some quadratic equations can be solved by factoring, others require more advanced methods. The method of completing the square is particularly useful because it works for all quadratic equations, whether their solutions are real or complex.

Understanding the Concept of Completing the Square

The core idea behind completing the square is to manipulate the equation so that one side becomes a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as $(x+d)^2 = x^2+2dx+d^2$. Our goal is to rewrite the given quadratic equation in a way that matches this pattern.

Let's consider the general form of a quadratic equation: $ax^2+bx+c=0$. To apply completing the square effectively, it's often easiest if the coefficient of the $x^2$ term, $a$, is 1. If it's not, we can divide the entire equation by $a$. In our specific equation, $x^2+4x+8=0$, the coefficient of $x^2$ is already 1, which simplifies the process.

Now, let's focus on the terms involving $x$. We have $x^2+4x$. We want to add a constant term to this expression to make it a perfect square trinomial. To find this constant, we take the coefficient of the $x$ term (which is 4), divide it by 2, and then square the result. So, $(4/2)^2 = 2^2 = 4$. This is the number we need to add to $x^2+4x$ to complete the square. If we add 4, the expression becomes $x^2+4x+4$, which is a perfect square trinomial that can be factored as $(x+2)^2$.

However, we can't just add a number to one side of an equation without affecting its balance. Whatever we do to one side, we must do to the other. Therefore, when we add 4 to complete the square on the left side of $x^2+4x+8=0$, we must also subtract 4 to maintain equality. This leads us to rewrite the equation. The method of completing the square is a systematic approach that transforms the equation into a more manageable form, ultimately revealing the roots of the quadratic equation.

Step-by-Step Solution Using Completing the Square

Let's begin by solving the equation $x^2+4x+8=0$ using the completing the square method. Our first step is to isolate the terms involving $x$ on one side of the equation. We can do this by moving the constant term (8) to the right side:

$x^2+4x = -8$

Now, we need to complete the square on the left side. As we discussed, we take the coefficient of the $x$ term (which is 4), divide it by 2, and square the result. $(4/2)^2 = 2^2 = 4$. This is the value we need to add to both sides of the equation to maintain balance:

$x^2+4x + 4 = -8 + 4$

On the left side, $x^2+4x+4$ is now a perfect square trinomial, which can be factored as $(x+2)^2$. On the right side, we simplify the constants: $-8+4 = -4$. So, our equation becomes:

$(x+2)^2 = -4$

Now that we have the equation in a squared form, we can solve for $x$ by taking the square root of both sides. Remember that when taking the square root, we must consider both the positive and negative roots:

$\sqrt{(x+2)^2} = \pm\sqrt{-4}$

$x+2 = \pm\sqrt{-4}$

Dealing with the square root of a negative number introduces complex numbers. We know that $\sqrt{-1} = i$, where $i$ is the imaginary unit. Therefore, $\sqrt{-4}$ can be rewritten as $\sqrt{4} imes \sqrt{-1} = 2i$.

So, our equation is now:

$x+2 = \pm 2i$

To find the values of $x$, we simply subtract 2 from both sides:

$x = -2 \pm 2i$

These are the two solutions to the quadratic equation $x^2+4x+8=0$. They are complex conjugates, a common occurrence when solving quadratic equations with a negative discriminant.

Analyzing the Solutions

Our solutions are $x = -2 + 2i$ and $x = -2 - 2i$. These are complex numbers, consisting of a real part (-2) and an imaginary part ($\pm 2i$). The presence of complex solutions indicates that the parabola represented by the equation $y = x^2+4x+8$ does not intersect the x-axis. If we were to graph this parabola, it would lie entirely above or below the x-axis.

Let's verify these solutions by plugging them back into the original equation. For $x = -2 + 2i$:

$(-2 + 2i)^2 + 4(-2 + 2i) + 8$

$= (4 - 8i + 4i^2) + (-8 + 8i) + 8$

Since $i^2 = -1$, we have:

$= (4 - 8i - 4) - 8 + 8i + 8$

$= (-8i) - 8 + 8i + 8$

$= 0$

Now, let's check for $x = -2 - 2i$:

$(-2 - 2i)^2 + 4(-2 - 2i) + 8$

$= (4 + 8i + 4i^2) + (-8 - 8i) + 8$

$= (4 + 8i - 4) - 8 - 8i + 8$

$= (8i) - 8 - 8i + 8$

$= 0$

Both solutions satisfy the original equation, confirming our calculations. The method of completing the square has successfully yielded the complex roots of this quadratic equation.

Conclusion

We have successfully solved the quadratic equation $x^2+4x+8=0$ by completing the square. This systematic method allowed us to transform the equation into a perfect square form, leading us to the solutions $x = -2 \pm 2i$. Understanding completing the square is a valuable skill for any mathematics student, as it provides a reliable way to solve all quadratic equations and deepens our understanding of their properties, including the nature of their roots.

For further exploration into quadratic equations and algebraic techniques, you might find resources from Khan Academy helpful. They offer a wide range of lessons and practice problems that can solidify your understanding of these fundamental mathematical concepts.