Problem 89 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The equation \((x+5)^{2}=8\) is equivalent to \(x+5=2 \sqrt{2}\).

Step-by-Step Solution

Answer

The statement 'The equation \((x+5)^{2}=8\) is equivalent to \(x+5=2 \sqrt{2}\)' is true. So, there is no need to make any changes to this statement. It is correct as it is.

Step 1: Understanding the given equations

We have two equations. The first one is \((x+5)^{2} = 8\) and the second one is \(x+5 = 2\sqrt{2}\). The main task is to determine if these two are equivalent or not.

Step 2: Simplify and transform the first equation

Let's examine the square in the first equation. The square in the equation \((x+5)^{2} = 8\) means that we have \((x+5)\) multiplied by itself, or \((x+5)(x+5) = 8\). Squaring it, we get \(x^2 + 10x + 25 = 8\). Now, let's subtract 8 from both sides of this equation to get \(x^2 + 10x + 17 = 0\). This is the simplified form of the first equation.

Step 3: Determine the roots of the simplified equation

To figure out if \(x+5 = 2\sqrt{2}\) is indeed a solution to the equation, we need to find its roots. However this equation, \(x^2 + 10x + 17 = 0\), is a quadratic equation, and its roots cannot be found easily without using the quadratic formula or factoring. However, we can still try to solve the second equation and verify its correctness by substituting its solution into the first equation.

Step 4: Isolate \(x\) in the second equation and verify

First, isolate \(x\) in the equation \(x + 5 = 2\sqrt{2}\) by subtracting 5 from both sides to get \(x = 2\sqrt{2} - 5\). Now substitute this \(x\) value into the original equation as follows: \(((2 \sqrt{2} - 5) + 5)^2\), which simplifies to \((2 \sqrt{2})^2\), and that is equal to 8. Thus, \(x + 5 = 2\sqrt{2}\) is indeed a correct solution for the first problem.

Key Concepts

Quadratic Equations

In mathematics, particularly algebra, quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They are called quadratic because the highest exponent of the variable \(x\) is 2. This type of equation can have different kinds of solutions, depending on the discriminant \(b^2 - 4ac\). If the discriminant is positive, there are two real and distinct solutions. If it's zero, there is exactly one real solution. If the discriminant is negative, the solutions are complex numbers.
Quadratic equations can appear in many different forms. For example, our exercise involves the equation \((x+5)^2 = 8\), where the quadratic term is hidden inside a squared expression. By expanding it, we see its true quadratic form as \(x^2 + 10x + 17 = 0\). Solving quadratic equations can involve several methods:

• Factoring, if the equation is easily factorable.
• Completing the square to make the expression into a perfect square trinomial.
• Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), which works for any quadratic equation and provides both solutions.

Each of these methods has its own advantages depending on the specific equation at hand.

Simplification

Simplification is an important step in solving any algebraic equation. It involves reducing the equation to its simplest possible form. With quadratic equations, simplification often starts with expanding expressions and combining like terms.
In our example, the original equation \((x+5)^2 = 8\) requires simplification. By expanding \((x+5)(x+5)\), we obtain \(x^2 + 10x + 25 = 8\). Once expanded, we simplify it further by subtracting 8 from both sides to get \(x^2 + 10x + 17 = 0\).

• When simplifying, ensure you correctly apply arithmetic operations and handle negative numbers carefully.
• Rearranging the terms helps in setting up the equation in a more recognizable form for solving.
• Simplification is not just about getting a smaller equation; it is also about preparing it for solution by making it more recognizable as a standard form.

With a simpler form, solving the equation becomes much more manageable.

Verifying Solutions

Verifying solutions is a crucial final step when solving equations to ensure they are correct. This process involves checking your solution by substituting it back into the original equation to see if it holds true.
In our exercise, once we simplified the original equation and solved for \(x\), we ended up with potential solutions. To verify if \((x+5) = 2\sqrt{2}\) is correct, we first isolate \(x\) to get \(x = 2\sqrt{2} - 5\). Substituting this value back into the original equation \((x+5)^2 = 8\) confirms the solution.

• Checking solutions helps catch errors made during simplification and calculation.
• It confirms that all arithmetic operations align with the original equation's requirements.
• Verification is especially important when dealing with complex equations involving multiple steps.

Making sure a solution is correct by verifying it strengthens understanding and guarantees correctness in your mathematical work.