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Simplifying (7x + 6) * 5x = 90 Reorder the terms: (6 + 7x) * 5x = 90 Reorder the terms for easier multiplication: 5x(6 + 7x) = 90 (6 * 5x + 7x * 5x) = 90 (30x + 35x2) = 90 Solving 30x + 35x2 = 90 Solving for variable 'x'. Reorder the terms: -90 + 30x + 35x2 = 90 + -90 Combine like terms: 90 + -90 = 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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