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Simplifying (7x + -1.5)(7x + -9.5) + (4x + 2)(4x + 1) = 0 Reorder the terms: (-1.5 + 7x)(7x + -9.5) + (4x + 2)(4x + 1) = 0 Reorder the terms: (-1.5 + 7x)(-9.5 + 7x) + (4x + 2)(4x + 1) = 0 Multiply (-1.5 + 7x) * (-9.5 + 7x) (-1.5(-9.5 + 7x) + 7x * (-9.5 + 7x)) + (4x + 2)(4x + 1) = 0 ((-9.5 * -1.5 + 7x * -1.5) + 7x * (-9.5 + 7x)) + (4x + 2)(4x + 1) = 0 ((14.25 + -10.5x) + 7x * (-9.5 + 7x)) + (4x + 2)(4x + 1) = 0 (14.25 + -10.5x + (-9.5 * 7x + 7x * 7x)) + (4x + 2)(4x + 1) = 0 (14.25 + -10.5x + (-66.5x + 49x2)) + (4x + 2)(4x + 1) = 0 Combine like terms: -10.5x + -66.5x = -77x (14.25 + -77x + 49x2) + (4x + 2)(4x + 1) = 0 Reorder the terms: 14.25 + -77x + 49x2 + (2 + 4x)(4x + 1) = 0 Reorder the terms: 14.25 + -77x + 49x2 + (2 + 4x)(1 + 4x) = 0 Multiply (2 + 4x) * (1 + 4x) 14.25 + -77x + 49x2 + (2(1 + 4x) + 4x * (1 + 4x)) = 0 14.25 + -77x + 49x2 + ((1 * 2 + 4x * 2) + 4x * (1 + 4x)) = 0 14.25 + -77x + 49x2 + ((2 + 8x) + 4x * (1 + 4x)) = 0 14.25 + -77x + 49x2 + (2 + 8x + (1 * 4x + 4x * 4x)) = 0 14.25 + -77x + 49x2 + (2 + 8x + (4x + 16x2)) = 0 Combine like terms: 8x + 4x = 12x 14.25 + -77x + 49x2 + (2 + 12x + 16x2) = 0 Reorder the terms: 14.25 + 2 + -77x + 12x + 49x2 + 16x2 = 0 Combine like terms: 14.25 + 2 = 16.25 16.25 + -77x + 12x + 49x2 + 16x2 = 0 Combine like terms: -77x + 12x = -65x 16.25 + -65x + 49x2 + 16x2 = 0 Combine like terms: 49x2 + 16x2 = 65x2 16.25 + -65x + 65x2 = 0 Solving 16.25 + -65x + 65x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 65 the coefficient of the squared term: Divide each side by '65'. 0.25 + -1x + x2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -1x + -0.25 + x2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -1x + x2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -1x + x2 = 0 + -0.25 -1x + x2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -1x + x2 = -0.25 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = -0.25 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.25 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.25 + -1x + x2 = 0.00 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.00 Calculate the square root of the right side: 0.00 Break this problem into two subproblems by setting (x + -0.5) equal to 0.00 and 0.00.Subproblem 1
x + -0.5 = 0.00 Simplifying x + -0.5 = 0.00 Reorder the terms: -0.5 + x = 0.00 Solving -0.5 + x = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.00 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.00 + 0.5 x = 0.00 + 0.5 Combine like terms: 0.00 + 0.5 = 0.5 x = 0.5 Simplifying x = 0.5Subproblem 2
x + -0.5 = 0.00 Simplifying x + -0.5 = 0.00 Reorder the terms: -0.5 + x = 0.00 Solving -0.5 + x = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.00 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.00 + 0.5 x = 0.00 + 0.5 Combine like terms: 0.00 + 0.5 = 0.5 x = 0.5 Simplifying x = 0.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5, 0.5}
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