Solving $x^2 - 6 = 16x + 30$: Standard Form & Solutions

In this article, we will walk you through the process of rewriting the quadratic equation $x^2 - 6 = 16x + 30$ in standard form, factoring it, and finding the values of $x$ that satisfy the equation. Understanding how to solve quadratic equations is a fundamental skill in algebra, and this guide will provide a clear and comprehensive explanation.

Understanding the Quadratic Equation

First, let's delve into the quadratic equation itself. Quadratic equations are polynomial equations 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, with $a$ not equal to 0.
• $x$ is the variable we want to solve for.

Our goal is to transform the given equation, $x^2 - 6 = 16x + 30$, into this standard form. Once we have it in standard form, we can then factor it and find the solutions for $x$.

Step 1: Rewriting in Standard Form

To rewrite the equation in standard form, we need to move all the terms to one side of the equation, leaving zero on the other side. Let's start by subtracting $16x$ and $30$ from both sides of the equation:

$x^2 - 6 - 16x - 30 = 16x + 30 - 16x - 30$

This simplifies to:

$x^2 - 16x - 36 = 0$

Now, the equation is in standard form, where $a = 1$, $b = -16$, and $c = -36$. Having the equation in this form is crucial for the next step: factoring.

Step 2: Factoring the Quadratic Equation

Factoring a quadratic equation involves expressing it as a product of two binomials. In other words, we want to find two expressions of the form $(x + p)$ and $(x + q)$ such that:

$(x + p)(x + q) = x^2 - 16x - 36$

To find the values of $p$ and $q$, we need to find two numbers that:

1. Multiply to $c$ (which is -36 in our case).
2. Add up to $b$ (which is -16 in our case).

Let's list the factor pairs of -36:

• 1 and -36
• -1 and 36
• 2 and -18
• -2 and 18
• 3 and -12
• -3 and 12
• 4 and -9
• -4 and 9
• 6 and -6

Looking at these pairs, we can see that the pair 2 and -18 satisfy our conditions:

• $2 imes -18 = -36$
• $2 + (-18) = -16$

Therefore, we can factor the quadratic equation as:

$(x + 2)(x - 18) = 0$

Step 3: Finding the Solutions for $x$

Now that we have factored the equation, we can find the values of $x$ that make the equation true. This is based on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

In our case, we have:

$(x + 2)(x - 18) = 0$

This means either $(x + 2) = 0$ or $(x - 18) = 0$ (or both).

Let's solve each of these equations separately:

1. $x + 2 = 0$

   Subtract 2 from both sides:

   $x = -2$

2. $x - 18 = 0$

   Add 18 to both sides:

   $x = 18$

So, the solutions for $x$ are -2 and 18. These are the values that, when substituted back into the original equation, will make the equation true.

Summary of Steps

Let's recap the steps we took to solve the quadratic equation:

1. Rewrite in Standard Form: Transformed the equation $x^2 - 6 = 16x + 30$ into $x^2 - 16x - 36 = 0$.
2. Factor the Quadratic Equation: Factored the equation as $(x + 2)(x - 18) = 0$.
3. Find the Solutions for $x$: Used the zero-product property to find the solutions $x = -2$ and $x = 18$.

Importance of Standard Form and Factoring

Rewriting a quadratic equation in standard form is a crucial first step because it allows us to easily identify the coefficients $a$, $b$, and $c$. These coefficients are essential for various methods of solving quadratic equations, including factoring, completing the square, and using the quadratic formula.

Factoring is a powerful technique for solving quadratic equations because it transforms the equation into a product of simpler expressions. The zero-product property then allows us to quickly find the solutions by setting each factor equal to zero.

Alternative Methods for Solving Quadratic Equations

While factoring is an efficient method when the quadratic equation can be easily factored, there are other methods available for solving quadratic equations, such as:

• Completing the Square: This method involves manipulating the equation to create a perfect square trinomial, which can then be easily solved.

• Quadratic Formula: The quadratic formula is a general formula that can be used to solve any quadratic equation, regardless of whether it can be factored easily. The formula is:

  x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}

  Where $a$, $b$, and $c$ are the coefficients of the quadratic equation in standard form.

The choice of method depends on the specific equation and personal preference. However, understanding factoring is essential as it often provides the quickest solution when applicable.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications in various fields, including:

• Physics: Projectile motion, where the path of an object thrown into the air can be modeled using a quadratic equation.
• Engineering: Designing bridges, arches, and other structures where quadratic equations are used to calculate stress and strain.
• Economics: Modeling supply and demand curves, where quadratic equations can help determine equilibrium prices and quantities.
• Computer Graphics: Creating curves and surfaces in 3D graphics using quadratic equations and other mathematical functions.

Understanding quadratic equations and how to solve them is therefore valuable not only in mathematics but also in many practical applications.

Conclusion

In this guide, we have demonstrated how to rewrite the equation $x^2 - 6 = 16x + 30$ in standard form, factor it, and find the solutions for $x$. We found that the equation in standard form is $x^2 - 16x - 36 = 0$, and it can be factored as $(x + 2)(x - 18) = 0$. The solutions for $x$ are -2 and 18.

Mastering the process of solving quadratic equations is crucial for anyone studying algebra and related fields. By understanding the steps involved and practicing different methods, you can confidently tackle a wide range of quadratic equation problems.

For further exploration and practice, consider visiting trusted mathematical resources such as Khan Academy's Quadratic Equations section. This resource provides comprehensive lessons, practice exercises, and videos to help you deepen your understanding of quadratic equations.