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Simplifying (7a + -15)(9a + 25) = 180 Reorder the terms: (-15 + 7a)(9a + 25) = 180 Reorder the terms: (-15 + 7a)(25 + 9a) = 180 Multiply (-15 + 7a) * (25 + 9a) (-15(25 + 9a) + 7a * (25 + 9a)) = 180 ((25 * -15 + 9a * -15) + 7a * (25 + 9a)) = 180 ((-375 + -135a) + 7a * (25 + 9a)) = 180 (-375 + -135a + (25 * 7a + 9a * 7a)) = 180 (-375 + -135a + (175a + 63a2)) = 180 Combine like terms: -135a + 175a = 40a (-375 + 40a + 63a2) = 180 Solving -375 + 40a + 63a2 = 180 Solving for variable 'a'. Reorder the terms: -375 + -180 + 40a + 63a2 = 180 + -180 Combine like terms: -375 + -180 = -555 -555 + 40a + 63a2 = 180 + -180 Combine like terms: 180 + -180 = 0 -555 + 40a + 63a2 = 0 Begin completing the square. Divide all terms by 63 the coefficient of the squared term: Divide each side by '63'. -8.80952381 + 0.6349206349a + a2 = 0 Move the constant term to the right: Add '8.80952381' to each side of the equation. -8.80952381 + 0.6349206349a + 8.80952381 + a2 = 0 + 8.80952381 Reorder the terms: -8.80952381 + 8.80952381 + 0.6349206349a + a2 = 0 + 8.80952381 Combine like terms: -8.80952381 + 8.80952381 = 0.00000000 0.00000000 + 0.6349206349a + a2 = 0 + 8.80952381 0.6349206349a + a2 = 0 + 8.80952381 Combine like terms: 0 + 8.80952381 = 8.80952381 0.6349206349a + a2 = 8.80952381 The a term is 0.6349206349a. Take half its coefficient (0.3174603175). Square it (0.1007810532) and add it to both sides. Add '0.1007810532' to each side of the equation. 0.6349206349a + 0.1007810532 + a2 = 8.80952381 + 0.1007810532 Reorder the terms: 0.1007810532 + 0.6349206349a + a2 = 8.80952381 + 0.1007810532 Combine like terms: 8.80952381 + 0.1007810532 = 8.9103048632 0.1007810532 + 0.6349206349a + a2 = 8.9103048632 Factor a perfect square on the left side: (a + 0.3174603175)(a + 0.3174603175) = 8.9103048632 Calculate the square root of the right side: 2.985013377 Break this problem into two subproblems by setting (a + 0.3174603175) equal to 2.985013377 and -2.985013377.Subproblem 1
a + 0.3174603175 = 2.985013377 Simplifying a + 0.3174603175 = 2.985013377 Reorder the terms: 0.3174603175 + a = 2.985013377 Solving 0.3174603175 + a = 2.985013377 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3174603175' to each side of the equation. 0.3174603175 + -0.3174603175 + a = 2.985013377 + -0.3174603175 Combine like terms: 0.3174603175 + -0.3174603175 = 0.0000000000 0.0000000000 + a = 2.985013377 + -0.3174603175 a = 2.985013377 + -0.3174603175 Combine like terms: 2.985013377 + -0.3174603175 = 2.6675530595 a = 2.6675530595 Simplifying a = 2.6675530595Subproblem 2
a + 0.3174603175 = -2.985013377 Simplifying a + 0.3174603175 = -2.985013377 Reorder the terms: 0.3174603175 + a = -2.985013377 Solving 0.3174603175 + a = -2.985013377 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3174603175' to each side of the equation. 0.3174603175 + -0.3174603175 + a = -2.985013377 + -0.3174603175 Combine like terms: 0.3174603175 + -0.3174603175 = 0.0000000000 0.0000000000 + a = -2.985013377 + -0.3174603175 a = -2.985013377 + -0.3174603175 Combine like terms: -2.985013377 + -0.3174603175 = -3.3024736945 a = -3.3024736945 Simplifying a = -3.3024736945Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.6675530595, -3.3024736945}
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