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Simplifying 12x2 + 25x = 84 Reorder the terms: 25x + 12x2 = 84 Solving 25x + 12x2 = 84 Solving for variable 'x'. Reorder the terms: -84 + 25x + 12x2 = 84 + -84 Combine like terms: 84 + -84 = 0 -84 + 25x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -7 + 2.083333333x + x2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 2.083333333x + 7 + x2 = 0 + 7 Reorder the terms: -7 + 7 + 2.083333333x + x2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 2.083333333x + x2 = 0 + 7 2.083333333x + x2 = 0 + 7 Combine like terms: 0 + 7 = 7 2.083333333x + x2 = 7 The x term is 2.083333333x. Take half its coefficient (1.041666667). Square it (1.085069445) and add it to both sides. Add '1.085069445' to each side of the equation. 2.083333333x + 1.085069445 + x2 = 7 + 1.085069445 Reorder the terms: 1.085069445 + 2.083333333x + x2 = 7 + 1.085069445 Combine like terms: 7 + 1.085069445 = 8.085069445 1.085069445 + 2.083333333x + x2 = 8.085069445 Factor a perfect square on the left side: (x + 1.041666667)(x + 1.041666667) = 8.085069445 Calculate the square root of the right side: 2.843425653 Break this problem into two subproblems by setting (x + 1.041666667) equal to 2.843425653 and -2.843425653.Subproblem 1
x + 1.041666667 = 2.843425653 Simplifying x + 1.041666667 = 2.843425653 Reorder the terms: 1.041666667 + x = 2.843425653 Solving 1.041666667 + x = 2.843425653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.041666667' to each side of the equation. 1.041666667 + -1.041666667 + x = 2.843425653 + -1.041666667 Combine like terms: 1.041666667 + -1.041666667 = 0.000000000 0.000000000 + x = 2.843425653 + -1.041666667 x = 2.843425653 + -1.041666667 Combine like terms: 2.843425653 + -1.041666667 = 1.801758986 x = 1.801758986 Simplifying x = 1.801758986Subproblem 2
x + 1.041666667 = -2.843425653 Simplifying x + 1.041666667 = -2.843425653 Reorder the terms: 1.041666667 + x = -2.843425653 Solving 1.041666667 + x = -2.843425653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.041666667' to each side of the equation. 1.041666667 + -1.041666667 + x = -2.843425653 + -1.041666667 Combine like terms: 1.041666667 + -1.041666667 = 0.000000000 0.000000000 + x = -2.843425653 + -1.041666667 x = -2.843425653 + -1.041666667 Combine like terms: -2.843425653 + -1.041666667 = -3.88509232 x = -3.88509232 Simplifying x = -3.88509232Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.801758986, -3.88509232}
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