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Simplifying 20x2 + -12x = 90 Reorder the terms: -12x + 20x2 = 90 Solving -12x + 20x2 = 90 Solving for variable 'x'. Reorder the terms: -90 + -12x + 20x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + -12x + 20x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-45 + -6x + 10x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-45 + -6x + 10x2)' equal to zero and attempt to solve: Simplifying -45 + -6x + 10x2 = 0 Solving -45 + -6x + 10x2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -4.5 + -0.6x + x2 = 0.0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + -0.6x + 4.5 + x2 = 0.0 + 4.5 Reorder the terms: -4.5 + 4.5 + -0.6x + x2 = 0.0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + -0.6x + x2 = 0.0 + 4.5 -0.6x + x2 = 0.0 + 4.5 Combine like terms: 0.0 + 4.5 = 4.5 -0.6x + x2 = 4.5 The x term is -0.6x. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6x + 0.09 + x2 = 4.5 + 0.09 Reorder the terms: 0.09 + -0.6x + x2 = 4.5 + 0.09 Combine like terms: 4.5 + 0.09 = 4.59 0.09 + -0.6x + x2 = 4.59 Factor a perfect square on the left side: (x + -0.3)(x + -0.3) = 4.59 Calculate the square root of the right side: 2.142428529 Break this problem into two subproblems by setting (x + -0.3) equal to 2.142428529 and -2.142428529.Subproblem 1
x + -0.3 = 2.142428529 Simplifying x + -0.3 = 2.142428529 Reorder the terms: -0.3 + x = 2.142428529 Solving -0.3 + x = 2.142428529 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = 2.142428529 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = 2.142428529 + 0.3 x = 2.142428529 + 0.3 Combine like terms: 2.142428529 + 0.3 = 2.442428529 x = 2.442428529 Simplifying x = 2.442428529Subproblem 2
x + -0.3 = -2.142428529 Simplifying x + -0.3 = -2.142428529 Reorder the terms: -0.3 + x = -2.142428529 Solving -0.3 + x = -2.142428529 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = -2.142428529 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = -2.142428529 + 0.3 x = -2.142428529 + 0.3 Combine like terms: -2.142428529 + 0.3 = -1.842428529 x = -1.842428529 Simplifying x = -1.842428529Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.442428529, -1.842428529}Solution
x = {2.442428529, -1.842428529}
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