Solving $x^2 + 10x + 10 = 0$ By Completing The Square

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Let's dive into solving the quadratic equation $x^2 + 10x + 10 = 0$ using the method of completing the square. This technique is a powerful tool for finding the roots of quadratic equations, and it's especially useful when factoring isn't straightforward. We'll break down each step to ensure you understand the process thoroughly. You will not only learn how to solve this specific equation, but also gain a deeper understanding of the method itself.

Understanding the Method of Completing the Square

Before we jump into the specifics, let’s understand the essence of completing the square. At its heart, this method transforms a quadratic equation into a perfect square trinomial, which is a trinomial that can be factored into the square of a binomial. This form makes it easier to isolate the variable x and find the solutions. The main idea is to manipulate the equation algebraically so that one side becomes a perfect square trinomial. This involves adding a constant term to both sides of the equation. Remember, what we do on one side, we must do on the other to maintain equality.

Think of a perfect square trinomial as something that can be written in the form $(x + a)^2$ or $(x - a)^2$. When expanded, these expressions look like $x^2 + 2ax + a^2$ and $x^2 - 2ax + a^2$, respectively. Our goal is to massage our given equation into this form. We will add a constant value to both sides of the equation to make it fit the perfect square trinomial format. This will then allow us to easily solve for x. Let's get started with the steps, breaking down each part to make it easy to follow.

Step-by-Step Solution

1. Start with the Quadratic Equation

Our equation is $x^2 + 10x + 10 = 0$. The first step in completing the square is to isolate the constant term. We want to move the '+10' to the right side of the equation. To do this, we subtract 10 from both sides:

$x^2 + 10x = -10$

This sets us up to create our perfect square trinomial on the left-hand side. The next step is crucial, where we determine what constant to add to both sides.

2. Determine the Constant to Add

Here's where the magic of completing the square happens. We need to find a number that, when added to the left side, will make it a perfect square trinomial. To do this, we take half of the coefficient of the x term (which is 10), square it, and add the result to both sides of the equation. So, the coefficient of our x term is 10. We take half of it, which is 5, and then square it: $5^2 = 25$. This is the constant we'll add to both sides.

3. Add the Constant to Both Sides

Now, we add 25 to both sides of the equation:

$x^2 + 10x + 25 = -10 + 25$

This simplifies to:

$x^2 + 10x + 25 = 15$

Notice that the left side is now a perfect square trinomial. This is what we aimed for.

4. Factor the Perfect Square Trinomial

The left side of the equation, $x^2 + 10x + 25$, can be factored into $(x + 5)^2$. This is because $(x + 5)(x + 5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25$. So, our equation now looks like:

$(x + 5)^2 = 15$

We've successfully transformed the equation into a form where we can easily solve for x.

5. Take the Square Root of Both Sides

To get rid of the square on the left side, we take the square root of both sides of the equation. Remember to consider both the positive and negative square roots:

$\sqrt{(x + 5)^2} = \pm\sqrt{15}$

This gives us:

$x + 5 = \pm\sqrt{15}$

6. Isolate x

To isolate x, we subtract 5 from both sides:

$x = -5 \pm \sqrt{15}$

This gives us two solutions for x:

• $x = -5 + \sqrt{15}$
• $x = -5 - \sqrt{15}$

These are the roots of the quadratic equation. They are real numbers since $\sqrt{15}$ is a real number.

Verifying Real Number Solutions

Our solutions are $x = -5 + \sqrt{15}$ and $x = -5 - \sqrt{15}$. Since the square root of 15 is a real number, both solutions are real numbers. We can approximate these values to get a better sense of their magnitude:

• $x \approx -5 + 3.87 \approx -1.13$
• $x \approx -5 - 3.87 \approx -8.87$

These are indeed real numbers, confirming that our equation has real number solutions. This verification step is essential to ensure the correctness of the solutions. By understanding the discriminant, we can determine the nature of the roots without explicitly solving the equation. In this case, the positive discriminant confirms the existence of two distinct real roots.

Alternative Method: Using the Quadratic Formula

To further validate our solutions, let's use the quadratic formula, a general method for solving quadratic equations of the form $ax^2 + bx + c = 0$. The quadratic formula is given by:

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

For our equation, $x^2 + 10x + 10 = 0$, we have $a = 1$, $b = 10$, and $c = 10$. Plugging these values into the quadratic formula, we get:

$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(10)}}{2(1)}$

$x = \frac{-10 \pm \sqrt{100 - 40}}{2}$

$x = \frac{-10 \pm \sqrt{60}}{2}$

$x = \frac{-10 \pm 2\sqrt{15}}{2}$

$x = -5 \pm \sqrt{15}$

These are the same solutions we obtained by completing the square, which confirms our results. The quadratic formula provides a straightforward way to find the roots of any quadratic equation, and it's a valuable tool to have in your mathematical arsenal. It is important to know different methods to solve the same equation, as it allows for cross-validation and a deeper understanding of the mathematical concepts involved.

Visualizing the Solutions

Another way to understand the solutions is to visualize the quadratic equation as a parabola. The graph of $y = x^2 + 10x + 10$ is a parabola that opens upwards. The solutions to the equation $x^2 + 10x + 10 = 0$ are the x-intercepts of this parabola, i.e., the points where the parabola intersects the x-axis. The two x-intercepts correspond to the two real solutions we found. Graphing the parabola provides a visual confirmation of the existence and approximate values of the roots.

This graphical representation helps to connect the algebraic solutions with a geometric interpretation. The x-intercepts represent the values of x for which the quadratic expression equals zero, and the shape of the parabola gives insights into the behavior of the quadratic function. Tools like graphing calculators or online graphing utilities can be used to visualize the parabola and verify the solutions.

Conclusion

We have successfully solved the quadratic equation $x^2 + 10x + 10 = 0$ by completing the square. The solutions are $x = -5 + \sqrt{15}$ and $x = -5 - \sqrt{15}$, which are real numbers. We verified our solutions using the quadratic formula and discussed how to visualize the solutions graphically. Mastering the method of completing the square is a valuable skill in algebra, and it provides a deeper understanding of quadratic equations and their solutions. Remember, practice is key to mastering this technique. Try solving more quadratic equations by completing the square to build your confidence and proficiency. The more you practice, the more intuitive the process will become.

For further learning on quadratic equations and completing the square, you can visit Khan Academy's Quadratic Equations Section.