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Simplifying 0 = -1.5x2 + 12x + -10.5 Reorder the terms: 0 = -10.5 + 12x + -1.5x2 Solving 0 = -10.5 + 12x + -1.5x2 Solving for variable 'x'. Combine like terms: 0 + 10.5 = 10.5 10.5 + -12x + 1.5x2 = -10.5 + 12x + -1.5x2 + 10.5 + -12x + 1.5x2 Reorder the terms: 10.5 + -12x + 1.5x2 = -10.5 + 10.5 + 12x + -12x + -1.5x2 + 1.5x2 Combine like terms: -10.5 + 10.5 = 0.0 10.5 + -12x + 1.5x2 = 0.0 + 12x + -12x + -1.5x2 + 1.5x2 10.5 + -12x + 1.5x2 = 12x + -12x + -1.5x2 + 1.5x2 Combine like terms: 12x + -12x = 0 10.5 + -12x + 1.5x2 = 0 + -1.5x2 + 1.5x2 10.5 + -12x + 1.5x2 = -1.5x2 + 1.5x2 Combine like terms: -1.5x2 + 1.5x2 = 0.0 10.5 + -12x + 1.5x2 = 0.0 Begin completing the square. Divide all terms by 1.5 the coefficient of the squared term: Divide each side by '1.5'. 7 + -8x + x2 = 0 Move the constant term to the right: Add '-7' to each side of the equation. 7 + -8x + -7 + x2 = 0 + -7 Reorder the terms: 7 + -7 + -8x + x2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -8x + x2 = 0 + -7 -8x + x2 = 0 + -7 Combine like terms: 0 + -7 = -7 -8x + x2 = -7 The x term is -8x. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8x + 16 + x2 = -7 + 16 Reorder the terms: 16 + -8x + x2 = -7 + 16 Combine like terms: -7 + 16 = 9 16 + -8x + x2 = 9 Factor a perfect square on the left side: (x + -4)(x + -4) = 9 Calculate the square root of the right side: 3 Break this problem into two subproblems by setting (x + -4) equal to 3 and -3.Subproblem 1
x + -4 = 3 Simplifying x + -4 = 3 Reorder the terms: -4 + x = 3 Solving -4 + x = 3 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = 3 + 4 Combine like terms: -4 + 4 = 0 0 + x = 3 + 4 x = 3 + 4 Combine like terms: 3 + 4 = 7 x = 7 Simplifying x = 7Subproblem 2
x + -4 = -3 Simplifying x + -4 = -3 Reorder the terms: -4 + x = -3 Solving -4 + x = -3 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = -3 + 4 Combine like terms: -4 + 4 = 0 0 + x = -3 + 4 x = -3 + 4 Combine like terms: -3 + 4 = 1 x = 1 Simplifying x = 1Solution
The solution to the problem is based on the solutions from the subproblems. x = {7, 1}
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