Solving 5(x+6)^2-34=-29: A Quadratic Equation Guide

by Henrik Larsen 52 views

Hey guys! Let's dive into solving a quadratic equation today. Quadratic equations might seem intimidating at first, but with a systematic approach, they become quite manageable. We're going to break down the equation $5(x+6)^2-34=-29$ step by step, so you can follow along and understand each part of the process. This guide will not only provide the solution but also explain the underlying principles, ensuring you can tackle similar problems with confidence. So, grab your pencils, and let's get started!

Understanding Quadratic Equations

Before we jump into the solution, it's essential to understand what quadratic equations are and their general form. Quadratic equations are polynomial equations of the second degree. This means the highest power of the variable (usually denoted as $x$) is 2. The general form of a quadratic equation is expressed as:

ax2+bx+c=0ax^2 + bx + c = 0

Where $a$, $b$, and $c$ are constants, and $a$ is not equal to 0. If $a$ were 0, the equation would become linear, not quadratic. Quadratic equations pop up in various fields, from physics to engineering, and even in everyday life when dealing with areas and curves. Now, to solve these equations, we often aim to find the values of $x$ that satisfy the equation, which are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, such as factoring, completing the square, and using the quadratic formula. Each method has its strengths, and the best approach often depends on the specific equation you're dealing with. In our case, we'll tackle the equation $5(x+6)^2-34=-29$ using a method that simplifies the equation before expanding it, making the process smoother and more intuitive.

Methods for Solving Quadratic Equations

There are several methods you can use to solve quadratic equations, and understanding these different approaches is crucial for becoming proficient in algebra. Let's briefly explore some of the common methods:

  1. Factoring: This method involves rewriting the quadratic expression as a product of two binomials. For example, $x^2 + 5x + 6$ can be factored as $(x+2)(x+3)$. Setting each factor to zero gives you the solutions for $x$. Factoring is quick and straightforward when the quadratic equation can be easily factored, but it's not always applicable, especially when the roots are not integers or rational numbers.

  2. Completing the Square: This method involves transforming the quadratic equation into a perfect square trinomial. By adding and subtracting a specific constant, you can rewrite the equation in the form $(x+p)^2 = q$, where $p$ and $q$ are constants. Taking the square root of both sides then allows you to solve for $x$. Completing the square is a powerful method that works for all quadratic equations, and it's also the basis for deriving the quadratic formula.

  3. Quadratic Formula: The quadratic formula is a universal method that can solve any quadratic equation. It is derived from the method of completing the square and is given by:

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

    where $a$, $b$, and $c$ are the coefficients from the standard form of the quadratic equation, $ax^2 + bx + c = 0$. The quadratic formula is especially useful when factoring is difficult or impossible, and it provides a direct way to find the solutions, even if they are complex numbers.

  4. Graphical Method: While not a direct algebraic method, graphing the quadratic equation $y = ax^2 + bx + c$ can visually show the solutions. The roots of the equation are the points where the parabola intersects the x-axis. This method is useful for visualizing the solutions and understanding the nature of the roots.

In our specific problem, we will use a combination of simplification and square root properties to solve the equation efficiently. Recognizing which method is best suited for a particular equation comes with practice, so don't hesitate to try different approaches to strengthen your understanding.

Solving $5(x+6)^2-34=-29$: A Step-by-Step Guide

Alright, let's get our hands dirty and solve the given equation:

5(x+6)2−34=−295(x+6)^2-34=-29

Our goal here is to isolate $x$ and find its possible values. We'll do this by following a series of algebraic steps, making sure to keep the equation balanced at each stage. This means whatever operation we perform on one side of the equation, we must also perform on the other side. Let's break it down:

Step 1: Isolate the Squared Term

The first thing we want to do is to isolate the term that contains the square, which in this case is $5(x+6)^2$. To do this, we need to get rid of the -34 on the left side of the equation. We can accomplish this by adding 34 to both sides:

5(x+6)2−34+34=−29+345(x+6)^2 - 34 + 34 = -29 + 34

This simplifies to:

5(x+6)2=55(x+6)^2 = 5

Step 2: Divide to Simplify

Now that we've isolated the squared term, we need to get rid of the coefficient 5 that's multiplying $(x+6)^2$. To do this, we'll divide both sides of the equation by 5:

5(x+6)25=55\frac{5(x+6)^2}{5} = \frac{5}{5}

This simplifies to:

(x+6)2=1(x+6)^2 = 1

Step 3: Take the Square Root

We're getting closer! Now we have $(x+6)^2$ isolated. To undo the square, we'll take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots:

(x+6)2=±1\sqrt{(x+6)^2} = \pm \sqrt{1}

This gives us:

x+6=±1x + 6 = \pm 1

Step 4: Solve for $x$

Now we have two separate equations to solve for $x$:

  1. x+6=1x + 6 = 1

  2. x+6=−1x + 6 = -1

Let's solve each one individually.

Solving $x + 6 = 1$

To solve for $x$, we subtract 6 from both sides:

x+6−6=1−6x + 6 - 6 = 1 - 6

This gives us:

x=−5x = -5

Solving $x + 6 = -1$

Similarly, we subtract 6 from both sides:

x+6−6=−1−6x + 6 - 6 = -1 - 6

This gives us:

x=−7x = -7

Step 5: State the Solutions

So, we've found two solutions for $x$: -5 and -7. These are the values of $x$ that make the original equation true. We can write our solutions as:

x=−5,−7x = -5, -7

And there you have it! We've successfully solved the quadratic equation $5(x+6)^2-34=-29$ for all values of $x$. Remember, the key is to break down the problem into manageable steps and apply the correct algebraic operations. Now, let's summarize our findings and discuss how to verify our solutions.

Verifying the Solutions

It's always a good practice to verify your solutions to ensure they are correct. To verify our solutions, we'll substitute each value of $x$ back into the original equation and see if it holds true. We found that $x = -5$ and $x = -7$ are our solutions. Let's check them:

Verifying $x = -5$

Substitute $x = -5$ into the original equation:

5((−5)+6)2−34=−295((-5)+6)^2 - 34 = -29

Simplify inside the parentheses:

5(1)2−34=−295(1)^2 - 34 = -29

Continue simplifying:

5(1)−34=−295(1) - 34 = -29

5−34=−295 - 34 = -29

−29=−29-29 = -29

Since the equation holds true, $x = -5$ is indeed a solution.

Verifying $x = -7$

Now, substitute $x = -7$ into the original equation:

5((−7)+6)2−34=−295((-7)+6)^2 - 34 = -29

Simplify inside the parentheses:

5(−1)2−34=−295(-1)^2 - 34 = -29

Continue simplifying:

5(1)−34=−295(1) - 34 = -29

5−34=−295 - 34 = -29

−29=−29-29 = -29

Again, the equation holds true, confirming that $x = -7$ is also a solution. Therefore, we have confidently verified that our solutions are correct.

Verifying solutions is an essential step in problem-solving, especially in mathematics. It not only ensures the accuracy of your answers but also deepens your understanding of the problem. By substituting the solutions back into the original equation, you're essentially retracing your steps and confirming that each operation was performed correctly. This practice builds confidence and reinforces your problem-solving skills. Always remember to check your solutions, guys, it's worth the extra effort!

Conclusion

Great job, guys! We've successfully solved the quadratic equation $5(x+6)^2-34=-29$ for all values of $x$. We found the solutions to be $x = -5$ and $x = -7$. Remember, the key to solving quadratic equations is to follow a systematic approach: isolate the squared term, take the square root (considering both positive and negative roots), and then solve for $x$. Always verify your solutions to ensure accuracy.

Solving quadratic equations is a fundamental skill in algebra, and it's a stepping stone to more advanced mathematical concepts. The methods and techniques we've discussed here can be applied to a wide range of problems, so keep practicing, and you'll become more proficient over time. Whether you're tackling complex problems in math class or applying these principles in real-world scenarios, mastering quadratic equations will undoubtedly be a valuable asset. Keep up the great work, and remember, every problem solved is a step forward in your mathematical journey!