Solving $x^2 + 14x + 17 = -96$ A Step-by-Step Guide

In this comprehensive guide, we will delve into the process of solving quadratic equations, specifically focusing on the equation $x^2 + 14x + 17 = -96$. Quadratic equations, which take the general form of $ax^2 + bx + c = 0$, are fundamental in mathematics and have widespread applications in various fields, including physics, engineering, and economics. Understanding how to solve them is a crucial skill for anyone pursuing studies or careers in these areas.

Understanding Quadratic Equations

Before we dive into the solution, let's first establish a firm understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, x) is 2. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation are called roots or zeros, and they represent the values of x that satisfy the equation. These roots can be real or complex numbers.

There are several methods for solving quadratic equations, each with its own strengths and weaknesses. Some common methods include:

1. Factoring: This method involves rewriting the quadratic expression as a product of two linear factors. It is often the quickest method when applicable, but it is not always possible to factor a quadratic expression easily.
2. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side, which can then be factored as a squared binomial. It is a more general method than factoring and can be used to solve any quadratic equation.
3. Quadratic Formula: This formula provides a direct solution for the roots of any quadratic equation, regardless of whether it can be factored or not. It is a powerful tool but can be more computationally intensive than other methods.

In this guide, we will primarily use the quadratic formula to solve the given equation, but we will also briefly discuss the other methods for context.

Transforming the Equation into Standard Form

The first step in solving the equation $x^2 + 14x + 17 = -96$ is to transform it into the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. To do this, we need to move all the terms to one side of the equation, leaving zero on the other side.

In our case, we have $x^2 + 14x + 17 = -96$. To get zero on the right side, we add 96 to both sides of the equation:

$$x^2 + 14x + 17 + 96 = -96 + 96$$

This simplifies to:

$$x^2 + 14x + 113 = 0$$

Now, our equation is in the standard form, where a = 1, b = 14, and c = 113. We can now proceed to solve for x using the quadratic formula.

Applying the Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It states that for an equation in the form $ax^2 + bx + c = 0$, the solutions for x are given by:

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

This formula might look intimidating at first, but it is simply a matter of plugging in the values of a, b, and c from our equation and simplifying. In our case, we have a = 1, b = 14, and c = 113. Let's substitute these values into the quadratic formula:

$$x = \frac{-14 \pm \sqrt{14^2 - 4(1)(113)}}{2(1)}$$

Now, we need to simplify the expression inside the square root:

$$14^2 = 196$$

$$4(1)(113) = 452$$

So, we have:

$$x = \frac{-14 \pm \sqrt{196 - 452}}{2}$$

$$x = \frac{-14 \pm \sqrt{-256}}{2}$$

Notice that we have a negative number inside the square root. This indicates that the roots of the equation are complex numbers. We know that the square root of -1 is defined as the imaginary unit, denoted by i. Therefore, we can rewrite the expression as:

$$x = \frac{-14 \pm \sqrt{256 \cdot -1}}{2}$$

$$x = \frac{-14 \pm \sqrt{256} \cdot \sqrt{-1}}{2}$$

$$x = \frac{-14 \pm 16i}{2}$$

Finally, we can simplify the expression by dividing both terms in the numerator by 2:

$$x = -7 \pm 8i$$

Therefore, the solutions to the equation $x^2 + 14x + 17 = -96$ are $x = -7 + 8i$ and $x = -7 - 8i$.

Understanding Complex Roots

As we saw in the previous section, the solutions to our quadratic equation are complex numbers. Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). The a part is called the real part, and the b part is called the imaginary part.

In our case, the roots are -7 + 8i and -7 - 8i. Both roots have a real part of -7 and imaginary parts of 8 and -8, respectively. The presence of complex roots indicates that the parabola represented by the quadratic equation does not intersect the x-axis in the real number plane.

Complex roots always come in conjugate pairs. This means that if a + bi is a root of a quadratic equation with real coefficients, then a - bi is also a root. This is evident in our solution, where the roots are -7 + 8i and -7 - 8i, which are complex conjugates of each other.

Understanding complex roots is crucial in many areas of mathematics and physics, particularly in fields dealing with oscillations, waves, and electrical circuits.

Alternative Methods for Solving Quadratic Equations

While we used the quadratic formula to solve the equation $x^2 + 14x + 17 = -96$, it's worth briefly discussing other methods for solving quadratic equations, namely factoring and completing the square.

Factoring

Factoring involves rewriting the quadratic expression as a product of two linear factors. For example, the equation $x^2 + 5x + 6 = 0$ can be factored as (x + 2)(x + 3) = 0. The roots are then found by setting each factor equal to zero, giving x = -2 and x = -3.

However, not all quadratic equations can be easily factored. In our case, the equation $x^2 + 14x + 113 = 0$ does not factor nicely using real numbers. This is because the discriminant (the expression inside the square root in the quadratic formula, $b^2 - 4ac$) is negative, indicating complex roots.

Completing the Square

Completing the square is a method that involves manipulating the equation to create a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored as a squared binomial, such as $x^2 + 2ax + a^2 = (x + a)^2$.

To complete the square for the equation $x^2 + 14x + 113 = 0$, we first move the constant term to the right side:

$$x^2 + 14x = -113$$

Next, we take half of the coefficient of the x term (which is 14), square it (which gives 49), and add it to both sides:

$$x^2 + 14x + 49 = -113 + 49$$

$$x^2 + 14x + 49 = -64$$

Now, the left side is a perfect square trinomial, which can be factored as:

$$(x + 7)^2 = -64$$

Taking the square root of both sides gives:

$$x + 7 = \pm \sqrt{-64}$$

$$x + 7 = \pm 8i$$

Finally, we subtract 7 from both sides to solve for x:

$$x = -7 \pm 8i$$

As we can see, completing the square gives us the same solutions as the quadratic formula. While completing the square is a more general method than factoring, it can be more time-consuming than the quadratic formula for equations with complex roots.

Conclusion

In this guide, we have walked through the process of solving the quadratic equation $x^2 + 14x + 17 = -96$. We began by transforming the equation into standard form and then applied the quadratic formula to find the roots. We discovered that the roots are complex numbers, which means the parabola represented by the equation does not intersect the x-axis in the real number plane.

We also discussed the concept of complex roots and how they come in conjugate pairs. Additionally, we briefly explored alternative methods for solving quadratic equations, such as factoring and completing the square.

Solving quadratic equations is a fundamental skill in mathematics with applications in various fields. By understanding the different methods and their strengths and weaknesses, you can effectively tackle a wide range of quadratic equations and gain a deeper appreciation for the power and beauty of mathematics.

Therefore, the correct answer to the equation $x^2 + 14x + 17 = -96$ is B. $x = -7 egin{matrix} \pm \end{matrix} 8i$.