4x^2-15x-97=0

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Solution for 4x^2-15x-97=0 equation:


Simplifying
4x2 + -15x + -97 = 0

Reorder the terms:
-97 + -15x + 4x2 = 0

Solving
-97 + -15x + 4x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-24.25 + -3.75x + x2 = 0

Move the constant term to the right:

Add '24.25' to each side of the equation.
-24.25 + -3.75x + 24.25 + x2 = 0 + 24.25

Reorder the terms:
-24.25 + 24.25 + -3.75x + x2 = 0 + 24.25

Combine like terms: -24.25 + 24.25 = 0.00
0.00 + -3.75x + x2 = 0 + 24.25
-3.75x + x2 = 0 + 24.25

Combine like terms: 0 + 24.25 = 24.25
-3.75x + x2 = 24.25

The x term is -3.75x.  Take half its coefficient (-1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
-3.75x + 3.515625 + x2 = 24.25 + 3.515625

Reorder the terms:
3.515625 + -3.75x + x2 = 24.25 + 3.515625

Combine like terms: 24.25 + 3.515625 = 27.765625
3.515625 + -3.75x + x2 = 27.765625

Factor a perfect square on the left side:
(x + -1.875)(x + -1.875) = 27.765625

Calculate the square root of the right side: 5.269309727

Break this problem into two subproblems by setting 
(x + -1.875) equal to 5.269309727 and -5.269309727.

Subproblem 1

x + -1.875 = 5.269309727 Simplifying x + -1.875 = 5.269309727 Reorder the terms: -1.875 + x = 5.269309727 Solving -1.875 + x = 5.269309727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + x = 5.269309727 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + x = 5.269309727 + 1.875 x = 5.269309727 + 1.875 Combine like terms: 5.269309727 + 1.875 = 7.144309727 x = 7.144309727 Simplifying x = 7.144309727

Subproblem 2

x + -1.875 = -5.269309727 Simplifying x + -1.875 = -5.269309727 Reorder the terms: -1.875 + x = -5.269309727 Solving -1.875 + x = -5.269309727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + x = -5.269309727 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + x = -5.269309727 + 1.875 x = -5.269309727 + 1.875 Combine like terms: -5.269309727 + 1.875 = -3.394309727 x = -3.394309727 Simplifying x = -3.394309727

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.144309727, -3.394309727}

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