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Simplifying (15 + X)(39 + X) = 510 Multiply (15 + X) * (39 + X) (15(39 + X) + X(39 + X)) = 510 ((39 * 15 + X * 15) + X(39 + X)) = 510 ((585 + 15X) + X(39 + X)) = 510 (585 + 15X + (39 * X + X * X)) = 510 (585 + 15X + (39X + X2)) = 510 Combine like terms: 15X + 39X = 54X (585 + 54X + X2) = 510 Solving 585 + 54X + X2 = 510 Solving for variable 'X'. Reorder the terms: 585 + -510 + 54X + X2 = 510 + -510 Combine like terms: 585 + -510 = 75 75 + 54X + X2 = 510 + -510 Combine like terms: 510 + -510 = 0 75 + 54X + X2 = 0 Begin completing the square. Move the constant term to the right: Add '-75' to each side of the equation. 75 + 54X + -75 + X2 = 0 + -75 Reorder the terms: 75 + -75 + 54X + X2 = 0 + -75 Combine like terms: 75 + -75 = 0 0 + 54X + X2 = 0 + -75 54X + X2 = 0 + -75 Combine like terms: 0 + -75 = -75 54X + X2 = -75 The X term is 54X. Take half its coefficient (27). Square it (729) and add it to both sides. Add '729' to each side of the equation. 54X + 729 + X2 = -75 + 729 Reorder the terms: 729 + 54X + X2 = -75 + 729 Combine like terms: -75 + 729 = 654 729 + 54X + X2 = 654 Factor a perfect square on the left side: (X + 27)(X + 27) = 654 Calculate the square root of the right side: 25.573423705 Break this problem into two subproblems by setting (X + 27) equal to 25.573423705 and -25.573423705.Subproblem 1
X + 27 = 25.573423705 Simplifying X + 27 = 25.573423705 Reorder the terms: 27 + X = 25.573423705 Solving 27 + X = 25.573423705 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-27' to each side of the equation. 27 + -27 + X = 25.573423705 + -27 Combine like terms: 27 + -27 = 0 0 + X = 25.573423705 + -27 X = 25.573423705 + -27 Combine like terms: 25.573423705 + -27 = -1.426576295 X = -1.426576295 Simplifying X = -1.426576295Subproblem 2
X + 27 = -25.573423705 Simplifying X + 27 = -25.573423705 Reorder the terms: 27 + X = -25.573423705 Solving 27 + X = -25.573423705 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-27' to each side of the equation. 27 + -27 + X = -25.573423705 + -27 Combine like terms: 27 + -27 = 0 0 + X = -25.573423705 + -27 X = -25.573423705 + -27 Combine like terms: -25.573423705 + -27 = -52.573423705 X = -52.573423705 Simplifying X = -52.573423705Solution
The solution to the problem is based on the solutions from the subproblems. X = {-1.426576295, -52.573423705}
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