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Simplifying (x + -10)(x + -9) = 78 Reorder the terms: (-10 + x)(x + -9) = 78 Reorder the terms: (-10 + x)(-9 + x) = 78 Multiply (-10 + x) * (-9 + x) (-10(-9 + x) + x(-9 + x)) = 78 ((-9 * -10 + x * -10) + x(-9 + x)) = 78 ((90 + -10x) + x(-9 + x)) = 78 (90 + -10x + (-9 * x + x * x)) = 78 (90 + -10x + (-9x + x2)) = 78 Combine like terms: -10x + -9x = -19x (90 + -19x + x2) = 78 Solving 90 + -19x + x2 = 78 Solving for variable 'x'. Reorder the terms: 90 + -78 + -19x + x2 = 78 + -78 Combine like terms: 90 + -78 = 12 12 + -19x + x2 = 78 + -78 Combine like terms: 78 + -78 = 0 12 + -19x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-12' to each side of the equation. 12 + -19x + -12 + x2 = 0 + -12 Reorder the terms: 12 + -12 + -19x + x2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + -19x + x2 = 0 + -12 -19x + x2 = 0 + -12 Combine like terms: 0 + -12 = -12 -19x + x2 = -12 The x term is -19x. Take half its coefficient (-9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. -19x + 90.25 + x2 = -12 + 90.25 Reorder the terms: 90.25 + -19x + x2 = -12 + 90.25 Combine like terms: -12 + 90.25 = 78.25 90.25 + -19x + x2 = 78.25 Factor a perfect square on the left side: (x + -9.5)(x + -9.5) = 78.25 Calculate the square root of the right side: 8.845903006 Break this problem into two subproblems by setting (x + -9.5) equal to 8.845903006 and -8.845903006.Subproblem 1
x + -9.5 = 8.845903006 Simplifying x + -9.5 = 8.845903006 Reorder the terms: -9.5 + x = 8.845903006 Solving -9.5 + x = 8.845903006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x = 8.845903006 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x = 8.845903006 + 9.5 x = 8.845903006 + 9.5 Combine like terms: 8.845903006 + 9.5 = 18.345903006 x = 18.345903006 Simplifying x = 18.345903006Subproblem 2
x + -9.5 = -8.845903006 Simplifying x + -9.5 = -8.845903006 Reorder the terms: -9.5 + x = -8.845903006 Solving -9.5 + x = -8.845903006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x = -8.845903006 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x = -8.845903006 + 9.5 x = -8.845903006 + 9.5 Combine like terms: -8.845903006 + 9.5 = 0.654096994 x = 0.654096994 Simplifying x = 0.654096994Solution
The solution to the problem is based on the solutions from the subproblems. x = {18.345903006, 0.654096994}
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