\( x^2 + (x+1)^2 = 145 \) → \( x^2 + x^2 + 2x + 1 = 145 \) → \( 2x^2 + 2x - 144 = 0 \).
["Solving the Equation ( x^2 + (x+1)^2 = 145 ) Step-by-Step", "Solving quadratic equations is a fundamental skill in algebra, and one common type arises when dealing with sums of squared terms. The equation\n[ x^2 + (x+1)^2 = 145 ]\nrepresents a practical example often encountered in math problems. Here’s a detailed, step-by-step breakdown of how to simplify, solve, and understand this equation.", "---", "### Step 1: Expand the Squared Term", "Start by expanding ( (x+1)^2 ) using the binomial formula:\n[\n(x + 1)^2 = x^2 + 2x + 1\n]\nSubstitute this into the original equation:\n[\nx^2 + (x^2 + 2x + 1) = 145\n]", "---", "### Step 2: Combine Like Terms", "Now combine the like terms on the left-hand side:\n[\nx^2 + x^2 + 2x + 1 = 145\n\Rightarrow 2x^2 + 2x + 1 = 145\n]", "---", "### Step 3: Move All Terms to One Side", "To write the equation in standard quadratic form, subtract 145 from both sides:\n[\n2x^2 + 2x + 1 - 145 = 0\n\Rightarrow 2x^2 + 2x - 144 = 0\n]", "---", "### Step 4: Simplify the Quadratic Equation", "Factor out the greatest common factor (GCF), which is 2:\n[\n2(x^2 + x - 72) = 0\n]\nSince the product equals zero, each factor must be zero:\n[\nx^2 + x - 72 = 0\n]", "---", "### Step 5: Solve the Quadratic Equation", "Now solve ( x^2 + x - 72 = 0 ). This can be factored or solved using the quadratic formula.", "#### Factoring approach:\nWe seek two numbers that multiply to ( -72 ) and add to ( 1 ).\nThose numbers are ( 9 ) and ( -8 ), since:\n[\n9 \ imes (-8) = -72, \quad 9 + (-8) = 1\n]\nThus:\n[\nx^2 + x - 72 = (x + 9)(x - 8) = 0\n]", "Set each factor equal to zero:\n[\nx + 9 = 0 \Rightarrow x = -9\n\quad \ ext{or} \quad x - 8 = 0 \Rightarrow x = 8\n]", "---", "### Alternative: Using the Quadratic Formula", "For completeness, use the quadratic formula on ( 2x^2 + 2x - 144 = 0 ):\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \ ext{where } a=2, b=2, c=-144\n]\nCompute the discriminant:\n[\nb^2 - 4ac = 2^2 - 4(2)(-144) = 4 + 1152 = 1156\n]\nNow take the square root:\n[\n\sqrt{1156} = 34\n]\nSo:\n[\nx = \frac{-2 \pm 34}{4}\n]\nCompute both solutions:\n[\nx = \frac{-2 + 34}{4} = \frac{32}{4} = 8\n\quad \ ext{and} \quad\nx = \frac{-2 - 34}{4} = \frac{-36}{4} = -9\n]", "---", "### Final Answer", "The solutions to the equation ( x^2 + (x+1)^2 = 145 ) are:\n[\n\boxed{x = 8 \quad} \ ext{and} \quad x = -9\n]", "---", "### Why This Problem Matters", "Equations of the form ( x^2 + (x + c)^2 = k ) naturally appear in geometry (e.g., distance formulas), optimization problems, and physics modeling. Understanding how to expand, simplify, and solve such quadratics equips you to tackle complex real-world scenarios.", "Keywords for SEO:\n- Solve quadratic equation\n- Solve ( x^2 + (x+1)^2 = 145 )\n- Step-by-step quadratic solving\n- Algebraic equation simplification\n- Quadratic formula application\n- Math practice problem", "---", "Try solving similar problems today—whether it's geometry, physics, or data analysis—introducing small variations in quadratic forms helps build strong algebra skills!"]
