Problem 62 - Algebra 1

Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$x^{2}-7=57$$

Step-by-Step Solution

Answer

The solution to the equation \(x^{2}-7=57\) is \(x = \pm 8\)

Step 1: Reorganize the equation

The first step involves reorganizing the equation to get the unknown variable, x on one side of the equation. This can be done by adding 7 to both sides of the equation. This will result in the equation: \(x^{2} = 57+7\)

Step 2: Simplify the equation

Combine like terms on the right side of the equation to simplify it which gives: \(x^{2} = 64\)

Step 3: Solve for x

Now, to get the value of x, take the square root of both sides of the equation. But, don't forget that when you take the square root you always get 2 values, one negative and one positive. So, it results in \(x = \pm \sqrt{64}\)

Step 4: Final Answer

Since \(\sqrt{64}\) equals 8, the final solution is \(x = \pm 8\)

Key Concepts

Understanding Square Roots

Square roots are an essential concept when solving quadratic equations, like the example equation \(x^2 - 7 = 57\). The square root of a number \(n\), denoted as \(\sqrt{n}\), is a value that, when multiplied by itself, gives \(n\). In other words, \(\sqrt{n} \times \sqrt{n} = n\). For our problem, after rearranging and simplifying the equation, we found \(x^2 = 64\). To solve for \(x\), we take the square root of both sides.

Important notes about square roots:

• Every positive number \(n\) has two square roots: a positive square root \(\sqrt{n}\) and a negative square root \(-\sqrt{n}\).
• For example, \(\sqrt{64} = 8\) and \(-\sqrt{64} = -8\), which leads to solutions \(x = 8\) and \(x = -8\).
• Remember, the solution is both the positive and negative square root values to cover all possible solutions for \(x\).

This aspect is crucial because quadratic equations often have two solutions. Understanding and applying square roots correctly allows one to solve these equations efficiently.

Achieving Integer Solutions

Finding integer solutions, when possible, can simplify expressions and make results easier to interpret. Integers are whole numbers that can be positive, negative, or zero. In the equation \(x^2 - 7 = 57\), after rearranging and simplifying, we solved for \(x^2 = 64\). Since 64 is a perfect square, it naturally produces integer solutions when taking the square root.

Considerations for integer solutions include:

• If \(x^2 = n\) and \(n\) is a perfect square (like 64), then \(x = \pm \sqrt{n}\) will result in integer solutions.
• If \(n\) were not a perfect square, the solutions would involve radical expressions instead of integers.
• Working with integers makes calculations straightforward, as there's no need to handle decimal places or complex radicals.

By identifying whether \(n\) is a perfect square, one can determine whether integer solutions exist, simplifying the problem-solving process.

Rearranging Equations Effectively

Rearranging equations is often a vital first step in solving algebraic problems. In our original exercise, rearranging the equation \(x^2 - 7 = 57\) involved moving all terms to one side to isolate \(x^2\). We added 7 to both sides to achieve this, forming \(x^2 = 64\). This step is crucial because it sets up the equation for easier manipulation and solution in subsequent steps.

Key tips for rearranging equations:

• Keep balance: what you do to one side of the equation, do to the other. This maintains equality.
• Focus on isolating the variable term: it's essential to get \(x\) or \(x^2\) by itself on one side.
• Create a plan: consider the sequence of operations needed to simplify and solve the equation.

By strategically rearranging the equation, we can make complex problems more manageable, paving a clear path to finding solutions.