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

["# Simplifying the Equation: ( x^2 + x^2 + 2x + 1 = 85 )", "Solving equations can feel overwhelming at first, but simplifying expressions is the key to making algebra easier—especially when dealing with quadratic forms. One classic equation that demonstrates this process is:", "[\nx^2 + x^2 + 2x + 1 = 85\n]", "In this article, we’ll walk through how to simplify and solve this equation step-by-step, turning complexity into clarity. Whether you're a student mastering algebra or a lifelong learner brushing up on fundamentals, this straightforward simplification will highlight powerful techniques for equation solving.", "---", "## Step 1: Combine Like Terms", "The first simplification step involves combining like terms on the left-hand side. Notice that ( x^2 ) appears twice:", "[\nx^2 + x^2 = 2x^2\n]", "So the equation now becomes:", "[\n2x^2 + 2x + 1 = 85\n]", "This reduces the original four-term expression to a manageable quadratic form with only two terms and a constant.", "---", "## Step 2: Move All Terms to One Side", "To prepare for solving, we must set the equation to zero:", "[\n2x^2 + 2x + 1 - 85 = 0\n]", "Simplify the constant:", "[\n2x^2 + 2x - 84 = 0\n]", "Now the equation is in standard quadratic form:\n[\nax^2 + bx + c = 0\n]\nwhere ( a = 2 ), ( b = 2 ), and ( c = -84 ).", "---", "## Step 3: Simplify by Dividing Through by the Greatest Common Factor", "The entire equation is divisible by 2:\n[\n\frac{2x^2 + 2x - 84}{2} = 0 \quad \Rightarrow \quad x^2 + x - 42 = 0\n]", "This simplified version makes it easier to factor or apply the quadratic formula.", "---", "## Step 4: Factor the Simplified Quadratic (if possible)", "We now focus on solving:\n[\nx^2 + x - 42 = 0\n]", "Look for two numbers that multiply to (-42) and add to (1) (the coefficient of (x)). These numbers are (7) and (-6):", "[\nx^2 + 7x - 6x - 42 = 0\n]", "Group terms:", "[\n(x^2 + 7x) - (6x + 42) = 0\n]\n[\nx(x + 7) - 6(x + 7) = 0\n]", "Factor out the common binomial:", "[\n(x + 7)(x - 6) = 0\n]", "---", "## Step 5: Apply the Zero Product Property", "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]", "---", "## Final Answer", "The simplified equation ( x^2 + x^2 + 2x + 1 = 85 ) reduces elegantly to:\n[\nx^2 + x - 42 = 0,\n]\nwhich factors neatly to ( (x + 7)(x - 6) = 0 ), giving two real solutions:\n[\nx = -7 \quad \ ext{and} \quad x = 6\n]", "---", "## Why Simplification Matters", "Turning a complex equation like ( x^2 + x^2 + 2x + 1 = 85 ) into its simplified form reveals underlying patterns:\n- Combining like terms reduces redundancy.\n- Moving constants to one side isolates the variable.\n- Factoring transforms a quadratic into linear components, making solutions immediate.", "Mastering these techniques builds a strong foundation for tackling more advanced math with confidence and clarity.", "---", "Whether you're graphing, solving, or just trying to understand — simplification is the bridge between confusion and comprehension. Keep practicing, and watch how seemingly complicated equations transform into clear, solvable forms.", "---", "Keywords: simplify quadratic equation, solve (x^2 + x^2 + 2x + 1 = 85), algebraic simplification, factoring quadratics, step-by-step equation solving, algebra fundamentals."]