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Simplifying 6x(10 + -6x) = 35 + -5(6x + 1) (10 * 6x + -6x * 6x) = 35 + -5(6x + 1) (60x + -36x2) = 35 + -5(6x + 1) Reorder the terms: 60x + -36x2 = 35 + -5(1 + 6x) 60x + -36x2 = 35 + (1 * -5 + 6x * -5) 60x + -36x2 = 35 + (-5 + -30x) Combine like terms: 35 + -5 = 30 60x + -36x2 = 30 + -30x Solving 60x + -36x2 = 30 + -30x Solving for variable 'x'. Reorder the terms: -30 + 60x + 30x + -36x2 = 30 + -30x + -30 + 30x Combine like terms: 60x + 30x = 90x -30 + 90x + -36x2 = 30 + -30x + -30 + 30x Reorder the terms: -30 + 90x + -36x2 = 30 + -30 + -30x + 30x Combine like terms: 30 + -30 = 0 -30 + 90x + -36x2 = 0 + -30x + 30x -30 + 90x + -36x2 = -30x + 30x Combine like terms: -30x + 30x = 0 -30 + 90x + -36x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-5 + 15x + -6x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-5 + 15x + -6x2)' equal to zero and attempt to solve: Simplifying -5 + 15x + -6x2 = 0 Solving -5 + 15x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. 0.8333333333 + -2.5x + x2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + -2.5x + -0.8333333333 + x2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + -2.5x + x2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + -2.5x + x2 = 0 + -0.8333333333 -2.5x + x2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 -2.5x + x2 = -0.8333333333 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = -0.8333333333 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -0.8333333333 + 1.5625 Combine like terms: -0.8333333333 + 1.5625 = 0.7291666667 1.5625 + -2.5x + x2 = 0.7291666667 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 0.7291666667 Calculate the square root of the right side: 0.853912564 Break this problem into two subproblems by setting (x + -1.25) equal to 0.853912564 and -0.853912564.Subproblem 1
x + -1.25 = 0.853912564 Simplifying x + -1.25 = 0.853912564 Reorder the terms: -1.25 + x = 0.853912564 Solving -1.25 + x = 0.853912564 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 0.853912564 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.853912564 + 1.25 x = 0.853912564 + 1.25 Combine like terms: 0.853912564 + 1.25 = 2.103912564 x = 2.103912564 Simplifying x = 2.103912564Subproblem 2
x + -1.25 = -0.853912564 Simplifying x + -1.25 = -0.853912564 Reorder the terms: -1.25 + x = -0.853912564 Solving -1.25 + x = -0.853912564 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -0.853912564 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.853912564 + 1.25 x = -0.853912564 + 1.25 Combine like terms: -0.853912564 + 1.25 = 0.396087436 x = 0.396087436 Simplifying x = 0.396087436Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.103912564, 0.396087436}Solution
x = {2.103912564, 0.396087436}
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