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Simplifying 90 = 5X(7X + 6) Reorder the terms: 90 = 5X(6 + 7X) 90 = (6 * 5X + 7X * 5X) 90 = (30X + 35X2) Solving 90 = 30X + 35X2 Solving for variable 'X'. Reorder the terms: 90 + -30X + -35X2 = 30X + -30X + 35X2 + -35X2 Combine like terms: 30X + -30X = 0 90 + -30X + -35X2 = 0 + 35X2 + -35X2 90 + -30X + -35X2 = 35X2 + -35X2 Combine like terms: 35X2 + -35X2 = 0 90 + -30X + -35X2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(18 + -6X + -7X2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(18 + -6X + -7X2)' equal to zero and attempt to solve: Simplifying 18 + -6X + -7X2 = 0 Solving 18 + -6X + -7X2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -2.571428571 + 0.8571428571X + X2 = 0 Move the constant term to the right: Add '2.571428571' to each side of the equation. -2.571428571 + 0.8571428571X + 2.571428571 + X2 = 0 + 2.571428571 Reorder the terms: -2.571428571 + 2.571428571 + 0.8571428571X + X2 = 0 + 2.571428571 Combine like terms: -2.571428571 + 2.571428571 = 0.000000000 0.000000000 + 0.8571428571X + X2 = 0 + 2.571428571 0.8571428571X + X2 = 0 + 2.571428571 Combine like terms: 0 + 2.571428571 = 2.571428571 0.8571428571X + X2 = 2.571428571 The X term is 0.8571428571X. Take half its coefficient (0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. 0.8571428571X + 0.1836734694 + X2 = 2.571428571 + 0.1836734694 Reorder the terms: 0.1836734694 + 0.8571428571X + X2 = 2.571428571 + 0.1836734694 Combine like terms: 2.571428571 + 0.1836734694 = 2.7551020404 0.1836734694 + 0.8571428571X + X2 = 2.7551020404 Factor a perfect square on the left side: (X + 0.4285714286)(X + 0.4285714286) = 2.7551020404 Calculate the square root of the right side: 1.659850005 Break this problem into two subproblems by setting (X + 0.4285714286) equal to 1.659850005 and -1.659850005.Subproblem 1
X + 0.4285714286 = 1.659850005 Simplifying X + 0.4285714286 = 1.659850005 Reorder the terms: 0.4285714286 + X = 1.659850005 Solving 0.4285714286 + X = 1.659850005 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + X = 1.659850005 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + X = 1.659850005 + -0.4285714286 X = 1.659850005 + -0.4285714286 Combine like terms: 1.659850005 + -0.4285714286 = 1.2312785764 X = 1.2312785764 Simplifying X = 1.2312785764Subproblem 2
X + 0.4285714286 = -1.659850005 Simplifying X + 0.4285714286 = -1.659850005 Reorder the terms: 0.4285714286 + X = -1.659850005 Solving 0.4285714286 + X = -1.659850005 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + X = -1.659850005 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + X = -1.659850005 + -0.4285714286 X = -1.659850005 + -0.4285714286 Combine like terms: -1.659850005 + -0.4285714286 = -2.0884214336 X = -2.0884214336 Simplifying X = -2.0884214336Solution
The solution to the problem is based on the solutions from the subproblems. X = {1.2312785764, -2.0884214336}Solution
X = {1.2312785764, -2.0884214336}
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