So, \( x^2 - 1 = 2^3 = 8 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3 \).
["Solving the Equation: ( x^2 - 1 = 8 ) Step-by-Step", "Solving quadratic equations is a fundamental skill in algebra, and understanding how to isolate variables step-by-step is crucial for students and math enthusiasts alike. One common type of equation involves isolating ( x^2 ) from a simpler expression—this article breaks down the solution to ( x^2 - 1 = 8 ) clearly and thoroughly.", "---", "### Understanding the Equation: ( x^2 - 1 = 8 )", "The equation ( x^2 - 1 = 8 ) combines a squared term and a constant. To solve for ( x ), our goal is to isolate ( x^2 ) on one side of the equation.", "---", "### Step 1: Add 1 to Both Sides", "We begin by eliminating the constant subtracted on the left-hand side:", "[\nx^2 - 1 = 8\n]", "Add 1 to both sides:", "[\nx^2 - 1 + 1 = 8 + 1\n]", "This simplifies to:", "[\nx^2 = 9\n]", "---", "### Step 2: Take the Square Root of Both Sides", "Since ( x^2 = 9 ), we apply the square root to both sides to solve for ( x ):", "[\nx = \pm \sqrt{9}\n]", "Which gives:", "[\nx = \pm 3\n]", "This means ( x = 3 ) or ( x = -3 ).", "---", "### Why This Works: Key Algebraic Principles", "- Inverse Operations: To isolate ( x^2 ), we undid subtraction using addition, preserving equation balance.\n- Square Root Property: When solving ( x^2 = a ), the solutions are ( x = \sqrt{a} ) and ( x = -\sqrt{a} ), hence the ( \pm ) symbol.", "---", "### Final Answer", "The solutions to the equation ( x^2 - 1 = 8 ) are:", "[\n\boxed{x = 3 \quad \ ext{or} \quad x = -3}\n]", "---", "### Applications and Practice Tips", "This type of quadratic equation appears in physics, engineering, and optimization problems. To master solving such equations:", "- Practice isolating squares step-by-step.\n- Always verify solutions by substituting back into the original equation.\n- Understand the geometric meaning: solutions represent ( x )-coordinates where a parabola intersects the horizontal line ( y = 8 ).", "Mastering these foundational steps will build confidence and precision in algebra and beyond.", "---", "Keywords: solve ( x^2 - 1 = 8 ), quadratic equations, isolate ( x^2 ), apply square root property, algebraic steps, math tutorial\nMeta description: Learn how to solve ( x^2 - 1 = 8 ) step-by-step with clear algebraic reasoning and verification. Ideal for algebra students and math learners."]
