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Simplifying (13x + -5)(13x + -5) = 78 Reorder the terms: (-5 + 13x)(13x + -5) = 78 Reorder the terms: (-5 + 13x)(-5 + 13x) = 78 Multiply (-5 + 13x) * (-5 + 13x) (-5(-5 + 13x) + 13x * (-5 + 13x)) = 78 ((-5 * -5 + 13x * -5) + 13x * (-5 + 13x)) = 78 ((25 + -65x) + 13x * (-5 + 13x)) = 78 (25 + -65x + (-5 * 13x + 13x * 13x)) = 78 (25 + -65x + (-65x + 169x2)) = 78 Combine like terms: -65x + -65x = -130x (25 + -130x + 169x2) = 78 Solving 25 + -130x + 169x2 = 78 Solving for variable 'x'. Reorder the terms: 25 + -78 + -130x + 169x2 = 78 + -78 Combine like terms: 25 + -78 = -53 -53 + -130x + 169x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -53 + -130x + 169x2 = 0 Begin completing the square. Divide all terms by 169 the coefficient of the squared term: Divide each side by '169'. -0.3136094675 + -0.7692307692x + x2 = 0 Move the constant term to the right: Add '0.3136094675' to each side of the equation. -0.3136094675 + -0.7692307692x + 0.3136094675 + x2 = 0 + 0.3136094675 Reorder the terms: -0.3136094675 + 0.3136094675 + -0.7692307692x + x2 = 0 + 0.3136094675 Combine like terms: -0.3136094675 + 0.3136094675 = 0.0000000000 0.0000000000 + -0.7692307692x + x2 = 0 + 0.3136094675 -0.7692307692x + x2 = 0 + 0.3136094675 Combine like terms: 0 + 0.3136094675 = 0.3136094675 -0.7692307692x + x2 = 0.3136094675 The x term is -0.7692307692x. Take half its coefficient (-0.3846153846). Square it (0.1479289941) and add it to both sides. Add '0.1479289941' to each side of the equation. -0.7692307692x + 0.1479289941 + x2 = 0.3136094675 + 0.1479289941 Reorder the terms: 0.1479289941 + -0.7692307692x + x2 = 0.3136094675 + 0.1479289941 Combine like terms: 0.3136094675 + 0.1479289941 = 0.4615384616 0.1479289941 + -0.7692307692x + x2 = 0.4615384616 Factor a perfect square on the left side: (x + -0.3846153846)(x + -0.3846153846) = 0.4615384616 Calculate the square root of the right side: 0.679366221 Break this problem into two subproblems by setting (x + -0.3846153846) equal to 0.679366221 and -0.679366221.Subproblem 1
x + -0.3846153846 = 0.679366221 Simplifying x + -0.3846153846 = 0.679366221 Reorder the terms: -0.3846153846 + x = 0.679366221 Solving -0.3846153846 + x = 0.679366221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3846153846' to each side of the equation. -0.3846153846 + 0.3846153846 + x = 0.679366221 + 0.3846153846 Combine like terms: -0.3846153846 + 0.3846153846 = 0.0000000000 0.0000000000 + x = 0.679366221 + 0.3846153846 x = 0.679366221 + 0.3846153846 Combine like terms: 0.679366221 + 0.3846153846 = 1.0639816056 x = 1.0639816056 Simplifying x = 1.0639816056Subproblem 2
x + -0.3846153846 = -0.679366221 Simplifying x + -0.3846153846 = -0.679366221 Reorder the terms: -0.3846153846 + x = -0.679366221 Solving -0.3846153846 + x = -0.679366221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3846153846' to each side of the equation. -0.3846153846 + 0.3846153846 + x = -0.679366221 + 0.3846153846 Combine like terms: -0.3846153846 + 0.3846153846 = 0.0000000000 0.0000000000 + x = -0.679366221 + 0.3846153846 x = -0.679366221 + 0.3846153846 Combine like terms: -0.679366221 + 0.3846153846 = -0.2947508364 x = -0.2947508364 Simplifying x = -0.2947508364Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.0639816056, -0.2947508364}
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