Problem 54 - Elementary Algebra

Question

\((x+1)^{2}=64\)

Step-by-Step Solution

Answer

x = 7 or x = -9

Step 1: Expand the Equation

Recognize that \((x + 1)^2 = 64\) is already a simplified equation. No expansion is needed in this case.

Step 2: Take the Square Root of Both Sides

To solve for \(x\), take the square root of both sides of the equation: \[\sqrt{(x + 1)^2} = \pm\sqrt{64}\] This simplifies to: \[x + 1 = \pm 8\]

Step 3: Solve for x

Now, solve for \(x\) by isolating the variable. This results in two equations: \[x + 1 = 8\] \[x + 1 = -8\] For the first equation: \[x = 8 - 1\] \[x = 7\] For the second equation: \[x = -8 - 1\] \[x = -9\]

Step 4: Write the Final Answer

The solutions to the equation are: \[x = 7\] and \[x = -9\]

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that indicate the equality of two expressions. These equations involve variables, which are symbols (like x or y) representing unknown values. The goal is to solve these equations by finding the values of the variables that make the equation true.

For example, in the given exercise \((x + 1)^2 = 64\), we need to find the value of x that satisfies this relationship. Here, the equation combines basic algebraic operations like addition and squaring.

Square Roots

Square roots are fundamental in solving quadratic equations. The square root of a number is a value which, when multiplied by itself, gives the original number. It’s represented by the symbol \(\sqrt{}\).

In the exercise, we have \((x + 1)^2 = 64\). Taking the square root of both sides helps in simplifying the equation:

\[\sqrt{(x + 1)^2} = \pm \sqrt{64}\]

This simplifies to:

\[x + 1 = \pm 8\]

Notice the \(\pm\) symbol. It represents both the positive and negative roots because both 8 and -8, when squared, equal 64. This step converts a quadratic equation into a simpler linear equation.

Isolating Variables

Isolating variables means rearranging an equation to get the variable by itself on one side of the equation. This is a key step in solving equations.

In our exercise, after taking the square root, we get two linear equations:

\[x + 1 = 8\]

\[x + 1 = -8\]

To isolate x, subtract 1 from both sides:

For \[x + 1 = 8\]:

\[x = 8 - 1\]

\[x = 7\]

For \[x + 1 = -8\]:

\[x = -8 - 1\]

\[x = -9\]

By isolating the variable x, we determine the values that satisfy the original equation.

Quadratic Solutions

Quadratic solutions involve finding the values of x that satisfy a quadratic equation, which is any equation structured as \((ax^2 + bx + c = 0)\). The given exercise, though initially different in form, can be handled with methods designed for quadratics.

The equation \((x + 1)^2 = 64\) simplifies through taking the square root into two linear equations, offering two solutions: \(x = 7\) and \(x = -9\).

This example demonstrates a key property of quadratic equations: they often have two distinct solutions. This is essential to understand when solving such equations as it helps ensure that all possible answers are considered.