\( 2x^2 + 2x + 1 = 85 \)

["# Solving ( 2x^2 + 2x + 1 = 85 ): Step-by-Step Guide", "Solving quadratic equations is a fundamental algebraic skill with applications across science, engineering, and everyday problem-solving. One common equation students and professionals encounter is:", "[\n2x^2 + 2x + 1 = 85\n]", "This article walks you through solving this quadratic equation step-by-step, explains key concepts, and provides insights into graphing and analyzing quadratic functions.", "---", "## Understanding the Equation", "The given equation is:", "[\n2x^2 + 2x + 1 = 85\n]", "This is a quadratic equation in standard form:", "[\nax^2 + bx + c = 0\n]", "But first, we must rearrange the equation into this standard form by moving all terms to one side.", "---", "## Step 1: Rearranging to Standard Quadratic Form", "Subtract 85 from both sides:", "[\n2x^2 + 2x + 1 - 85 = 0\n]", "Simplify:", "[\n2x^2 + 2x - 84 = 0\n]", "Now in standard form:", "[\n2x^2 + 2x - 84 = 0\n]", "---", "## Step 2: Simplifying the Equation", "Before solving, simplify the equation by dividing every term by the greatest common divisor (GCD) of the coefficients.", "Here, all coefficients (2, 2, -84) are divisible by 2:", "[\n\frac{2x^2}{2} + \frac{2x}{2} + \frac{-84}{2} = 0\n]", "Simplifies to:", "[\nx^2 + x - 42 = 0\n]", "---", "## Step 3: Solving the Simplified Quadratic Equation", "Now solve:", "[\nx^2 + x - 42 = 0\n]", "### Option A: Factoring", "Look for two numbers that multiply to (-42) and add to (1).", "Factors of (-42):", "- (7) and (-6) → (7 \ imes (-6) = -42), (7 + (-6) = 1)", "Yes! So:", "[\nx^2 + x - 42 = (x + 7)(x - 6) = 0\n]", "Set each factor equal to zero:", "[\nx + 7 = 0 \quad \Rightarrow \quad x = -7\n]\n[\nx - 6 = 0 \quad \Rightarrow \quad x = 6\n]", "### Option B: Quadratic Formula (Verification)", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For (x^2 + x - 42 = 0), coefficients:\n(a = 1), (b = 1), (c = -42)", "Calculate discriminant:", "[\nb^2 - 4ac = (1)^2 - 4(1)(-42) = 1 + 168 = 169\n]", "Square root of discriminant:", "[\n\sqrt{169} = 13\n]", "Substitute:", "[\nx = \frac{-1 \pm 13}{2}\n]", "Two solutions:", "[\nx = \frac{-1 + 13}{2} = \frac{12}{2} = 6\n]\n[\nx = \frac{-1 - 13}{2} = \frac{-14}{2} = -7\n]", "Consistent results—confirmed: solutions are ( x = 6 ) and ( x = -7 ).", "---", "## Step 4: Graphing the Quadratic Function", "The original equation ( 2x^2 + 2x + 1 = 85 ) graphs as a parabola opening upward (since coefficient of (x^2) is positive).", "- The vertex lies midway between the roots:\n[\n x = \frac{-b}{2a} = \frac{-1}{2(2)} = -\frac{1}{4} = -0.25\n ]", "- Plug (x = -0.25) into the function to find the minimum (y)-value:", "[\n y = 2(-0.25)^2 + 2(-0.25) + 1 = 2(0.0625) - 0.5 + 1 = 0.125 - 0.5 + 1 = 0.625\n ]", "Wait—this contradicts earlier simplification. Let’s reconcile:", "Actually, after simplifying to (x^2 + x - 42 = 0), we solved for (x), but the vertex of (2x^2 + 2x - 84 = 0) is:", "[\nx = -\frac{2}{2 \cdot 2} = -\frac{1}{2} = -0.5\n]", "Plug into simplified function:", "[\nf(-0.5) = 2(-0.5)^2 + 2(-0.5) + 1 = 2(0.25) - 1 + 1 = 0.5 - 1 + 1 = 0.5\n]", "So the minimum point is at ( (-0.5, 0.5) ).", "This shows that while the equation yields two real solutions, the parabola’s deepest point lies above the (x = 0) threshold.", "---", "## Step 5: Real-World Applications", "Quadratic equations like (2x^2 + 2x + 1 = 85) appear in:", "- Physics: projectile motion and velocity problems\n- Economics: profit maximization models\n- Engineering: structural design and load calculations\n- Geometry: optimization of areas and volumes", "Understanding these roots helps predict real-world outcomes, from height over time to cost efficiency.", "---", "## Summary", "To solve ( 2x^2 + 2x + 1 = 85 ):", "1. Rearrange to ( 2x^2 + 2x - 84 = 0 )\n2. Simplify by dividing by 2: ( x^2 + x - 42 = 0 )\n3. Factor: ( (x + 7)(x - 6) = 0 )\n4. Solve: ( x = -7 ) and ( x = 6 )", "Use factoring for quick solutions, but verify with the quadratic formula. Always interpret results in context—especially when modeling real phenomena.", "---", "## Frequently Asked Questions (FAQ)", "Q: Why do I need to move all terms to one side?\nA: Quadratic equations are defined as ( ax^2 + bx + c = 0 ) so the quadratic formula and factoring methods apply correctly.", "Q: How do I verify my solutions?\nA: Substitute (x = 6) and (x = -7) back into the original equation—they should both satisfy it.", "Q: What does (x = -7) represent graphically?\nA: The points where the parabola intersects (y = 85), known as the roots or solution set.", "Q: Can less-than-zero values occur?\nA: Yes—quadratic models yield negative solutions depending on context (e.g., time, distance).", "---", "## Final Thoughts", "Mastering quadratic equations like (2x^2 + 2x + 1 = 85) builds both algebraic fluency and logical reasoning. Use this step-by-step guide to confidently solve equations, interpret their meaning, and apply them in varied fields.", "---", "Keywords:\n(2x^2 + 2x + 1 = 85), quadratic equation solution, factoring method, quadratic formula, solving quadratics, algebraic equation solving, real-world applications of quadratics, step-by-step math tutorial", "Meta Description:\nLearn how to solve (2x^2 + 2x + 1 = 85) step-by-step—factoring, quadratic formula, and graphing insights. Perfect for students and learners mastering algebra."]