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Simplifying -16x2 + 24x + 7 = 10 Reorder the terms: 7 + 24x + -16x2 = 10 Solving 7 + 24x + -16x2 = 10 Solving for variable 'x'. Reorder the terms: 7 + -10 + 24x + -16x2 = 10 + -10 Combine like terms: 7 + -10 = -3 -3 + 24x + -16x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -3 + 24x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.1875 + -1.5x + x2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -1.5x + -0.1875 + x2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -1.5x + x2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -1.5x + x2 = 0 + -0.1875 -1.5x + x2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -1.5x + x2 = -0.1875 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = -0.1875 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = -0.1875 + 0.5625 Combine like terms: -0.1875 + 0.5625 = 0.375 0.5625 + -1.5x + x2 = 0.375 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (x + -0.75) equal to 0.612372436 and -0.612372436.Subproblem 1
x + -0.75 = 0.612372436 Simplifying x + -0.75 = 0.612372436 Reorder the terms: -0.75 + x = 0.612372436 Solving -0.75 + x = 0.612372436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 0.612372436 + 0.75 x = 0.612372436 + 0.75 Combine like terms: 0.612372436 + 0.75 = 1.362372436 x = 1.362372436 Simplifying x = 1.362372436Subproblem 2
x + -0.75 = -0.612372436 Simplifying x + -0.75 = -0.612372436 Reorder the terms: -0.75 + x = -0.612372436 Solving -0.75 + x = -0.612372436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -0.612372436 + 0.75 x = -0.612372436 + 0.75 Combine like terms: -0.612372436 + 0.75 = 0.137627564 x = 0.137627564 Simplifying x = 0.137627564Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.362372436, 0.137627564}
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