\[ x(x - 2) = 2^3 = 8 \] - United Radiology

["Solving the Equation ( x(x - 2) = 8 ): A Step-by-Step Guide", "If you're tackling algebraic equations, one common challenge is solving quadratic forms like ( x(x - 2) = 8 ). In this article, we’ll walk through how to solve the equation ( x(x - 2) = 8 ), explain its key steps clearly, and highlight its importance in algebra and real-world applications.", "---", "### Understanding the Equation: ( x(x - 2) = 8 )", "The equation ( x(x - 2) = 8 ) is a quadratic equation written in standard form after expansion. It models a real situation where a quadratic relationship exists — such as area comparisons, projectile motion speeds, or financial growth scenarios.", "---", "### Step 1: Expand the Equation", "Start by distributing the left-hand side:", "[
\nx(x - 2) = x^2 - 2x
\n]", "So the equation becomes:", "[
\nx^2 - 2x = 8
\n]", "---", "### Step 2: Rearrange into Standard Quadratic Form", "Subtract 8 from both sides to set the equation to zero:", "[
\nx^2 - 2x - 8 = 0
\n]", "Now we have a standard quadratic equation:
\n[
\nx^2 - 2x - 8 = 0
\n]", "---", "### Step 3: Solve Using Factoring, Completing the Square, or the Quadratic Formula", "This trinomial can be solved directly by factoring:", "We look for two numbers that multiply to (-8) and add to (-2). These numbers are (-4) and (+2):", "[
\n(x - 4)(x + 2) = 0
\n]", "Set each factor equal to zero:", "[
\nx - 4 = 0 \quad \Rightarrow \quad x = 4
\n]
\n[
\nx + 2 = 0 \quad \Rightarrow \quad x = -2
\n]", "---", "### Verification: Plug back into original equation", "Check both solutions in ( x(x - 2) = 8 ):", "- For ( x = 4 ):
\n ( 4(4 - 2) = 4 \cdot 2 = 8 ) ✅
\n- For ( x = -2 ):
\n ( -2(-2 - 2) = -2 \cdot (-4) = 8 ) ✅", "Both solutions are valid.", "---", "### Why This Equation Matters", "Equations like ( x(x - 2) = 8 ) aren’t just abstract puzzles — they appear in:", "- Area calculations: Where dimensions differ by a constant and total area is known.
\n- Financial models: Showing profit vs. variable sales quantities.
\n- Physics: Estimating initial velocities from distance-time relationships.", "---", "### Final Answer", "The solutions to ( x(x - 2) = 8 ) are:
\n[
\n\boxed{x = 4 \quad} \ ext{and} \quad \boxed{x = -2}
\n]", "---", "If you're learning algebra, mastering such equations strengthens your skills in quadratic reasoning — essential for solving complex problems in math, science, engineering, and economics.", "Keywords: solve ( x(x - 2) = 8 ), quadratic equation steps, algebra solution, factoring quadratic, real-world equations, math practice.", "---", "Need more help solving equations? Try expanding, rearranging, and applying the quadratic formula for any quadratic—this approach works every time!"]