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Simplifying 6x2 + 19x + 7 = 90 Reorder the terms: 7 + 19x + 6x2 = 90 Solving 7 + 19x + 6x2 = 90 Solving for variable 'x'. Reorder the terms: 7 + -90 + 19x + 6x2 = 90 + -90 Combine like terms: 7 + -90 = -83 -83 + 19x + 6x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -83 + 19x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -13.83333333 + 3.166666667x + x2 = 0 Move the constant term to the right: Add '13.83333333' to each side of the equation. -13.83333333 + 3.166666667x + 13.83333333 + x2 = 0 + 13.83333333 Reorder the terms: -13.83333333 + 13.83333333 + 3.166666667x + x2 = 0 + 13.83333333 Combine like terms: -13.83333333 + 13.83333333 = 0.00000000 0.00000000 + 3.166666667x + x2 = 0 + 13.83333333 3.166666667x + x2 = 0 + 13.83333333 Combine like terms: 0 + 13.83333333 = 13.83333333 3.166666667x + x2 = 13.83333333 The x term is 3.166666667x. Take half its coefficient (1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. 3.166666667x + 2.506944447 + x2 = 13.83333333 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667x + x2 = 13.83333333 + 2.506944447 Combine like terms: 13.83333333 + 2.506944447 = 16.340277777 2.506944447 + 3.166666667x + x2 = 16.340277777 Factor a perfect square on the left side: (x + 1.583333334)(x + 1.583333334) = 16.340277777 Calculate the square root of the right side: 4.042310945 Break this problem into two subproblems by setting (x + 1.583333334) equal to 4.042310945 and -4.042310945.Subproblem 1
x + 1.583333334 = 4.042310945 Simplifying x + 1.583333334 = 4.042310945 Reorder the terms: 1.583333334 + x = 4.042310945 Solving 1.583333334 + x = 4.042310945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + x = 4.042310945 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = 4.042310945 + -1.583333334 x = 4.042310945 + -1.583333334 Combine like terms: 4.042310945 + -1.583333334 = 2.458977611 x = 2.458977611 Simplifying x = 2.458977611Subproblem 2
x + 1.583333334 = -4.042310945 Simplifying x + 1.583333334 = -4.042310945 Reorder the terms: 1.583333334 + x = -4.042310945 Solving 1.583333334 + x = -4.042310945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + x = -4.042310945 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = -4.042310945 + -1.583333334 x = -4.042310945 + -1.583333334 Combine like terms: -4.042310945 + -1.583333334 = -5.625644279 x = -5.625644279 Simplifying x = -5.625644279Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.458977611, -5.625644279}
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