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Simplifying 4x2 + 7x + 76 = 180 Reorder the terms: 76 + 7x + 4x2 = 180 Solving 76 + 7x + 4x2 = 180 Solving for variable 'x'. Reorder the terms: 76 + -180 + 7x + 4x2 = 180 + -180 Combine like terms: 76 + -180 = -104 -104 + 7x + 4x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -104 + 7x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -26 + 1.75x + x2 = 0 Move the constant term to the right: Add '26' to each side of the equation. -26 + 1.75x + 26 + x2 = 0 + 26 Reorder the terms: -26 + 26 + 1.75x + x2 = 0 + 26 Combine like terms: -26 + 26 = 0 0 + 1.75x + x2 = 0 + 26 1.75x + x2 = 0 + 26 Combine like terms: 0 + 26 = 26 1.75x + x2 = 26 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = 26 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = 26 + 0.765625 Combine like terms: 26 + 0.765625 = 26.765625 0.765625 + 1.75x + x2 = 26.765625 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 26.765625 Calculate the square root of the right side: 5.173550522 Break this problem into two subproblems by setting (x + 0.875) equal to 5.173550522 and -5.173550522.Subproblem 1
x + 0.875 = 5.173550522 Simplifying x + 0.875 = 5.173550522 Reorder the terms: 0.875 + x = 5.173550522 Solving 0.875 + x = 5.173550522 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 5.173550522 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 5.173550522 + -0.875 x = 5.173550522 + -0.875 Combine like terms: 5.173550522 + -0.875 = 4.298550522 x = 4.298550522 Simplifying x = 4.298550522Subproblem 2
x + 0.875 = -5.173550522 Simplifying x + 0.875 = -5.173550522 Reorder the terms: 0.875 + x = -5.173550522 Solving 0.875 + x = -5.173550522 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -5.173550522 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -5.173550522 + -0.875 x = -5.173550522 + -0.875 Combine like terms: -5.173550522 + -0.875 = -6.048550522 x = -6.048550522 Simplifying x = -6.048550522Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.298550522, -6.048550522}
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