Problem 27 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}=256 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -16\) and \(x = 16\).

Step 1: Recognize the equation type

Recognize that this is a simple quadratic equation, in which the square of a variable is equal to a constant number. In this case, the equation is \(x^{2} = 256\).

Step 2: Take the square root

To undo the squaring operation, take the square root of both sides of the equation to solve for \(x\). Remember, when we take the square root of both sides, we must include both the positive and negative roots. So, the equation becomes: \(x = \pm \sqrt{256}\).

Step 3: Calculate the value

Simplify the right side by calculating the square root of 256, which gives us 16. Due to the plus/minus sign in front of the square root, the solution to the equation is \(x = -16\) and \(x = 16\).

Key Concepts

Solving Equations

When faced with an equation, particularly a quadratic one like \( x^2 = 256 \), the primary goal is to determine the value of the unknown variable, in this case \( x \). Quadratic equations typically involve terms where the variable is squared.
The first step in solving such an equation is to identify its type. Recognizing a quadratic equation is crucial because it tells us that the solution involves finding the roots of the polynomial.

• Identify the equation: Look for expressions like \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.
• Isolate the variable term: For \( x^2 = 256 \), it's already in a form where the term involving \( x \) is isolated.

After recognizing that \( x^2 = 256 \) is a perfect square form, we can proceed to apply the square root to both sides for a direct solution.

Square Root

The square root is a mathematical operation that essentially "undoes" squaring. To solve \( x^2 = 256 \), we apply the square root to both sides of the equation. This involves a few key steps.

• Understanding square roots: The square root of a number \( n \) is a value that, when multiplied by itself, gives \( n \). Thus, \( \sqrt{n} \times \sqrt{n} = n \).
• Applying square roots: When you take the square root of both sides of \( x^2 = 256 \), you must consider both the positive and negative solutions. This is because both \( 16 \times 16 \) and \( -16 \times -16 \) equal 256.
• Simplifying the result: Once \( \sqrt{256} \) is computed as 16, we know that the solutions for \( x \) are \( x = 16 \) and \( x = -16 \).

These solutions demonstrate why understanding square roots is essential, particularly their applications in equations that include squared terms.

Integer Solutions

Integer solutions are solutions to an equation that are whole numbers without fractions or decimals. In the case of solving \( x^2 = 256 \), we identify if the solutions are integers.

• Verifying integer solutions: Here, \( x = 16 \) and \( x = -16 \) are both integers. They satisfy the equation perfectly as integer squares.
• Recognizing when integers occur: In cases where the squared constant, like 256, is a perfect square, the roots will often be integers. Perfect squares have integer square roots.
• Handling non-integer solutions: If square roots aren't perfect, solutions may involve radicals. For \( x^2 = 256 \), since 256 is a perfect square, the problem specifically yields integer solutions, simplifying verification and interpretation.

Understanding when and why an equation results in integer solutions is invaluable in simplifying mathematical problems and ensuring correct interpretation of results.