4x^2-7x-21=0

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Solution for 4x^2-7x-21=0 equation:


Simplifying
4x2 + -7x + -21 = 0

Reorder the terms:
-21 + -7x + 4x2 = 0

Solving
-21 + -7x + 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'.
-5.25 + -1.75x + x2 = 0

Move the constant term to the right:

Add '5.25' to each side of the equation.
-5.25 + -1.75x + 5.25 + x2 = 0 + 5.25

Reorder the terms:
-5.25 + 5.25 + -1.75x + x2 = 0 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + -1.75x + x2 = 0 + 5.25
-1.75x + x2 = 0 + 5.25

Combine like terms: 0 + 5.25 = 5.25
-1.75x + x2 = 5.25

The x term is -1.75x.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75x + 0.765625 + x2 = 5.25 + 0.765625

Reorder the terms:
0.765625 + -1.75x + x2 = 5.25 + 0.765625

Combine like terms: 5.25 + 0.765625 = 6.015625
0.765625 + -1.75x + x2 = 6.015625

Factor a perfect square on the left side:
(x + -0.875)(x + -0.875) = 6.015625

Calculate the square root of the right side: 2.452677109

Break this problem into two subproblems by setting 
(x + -0.875) equal to 2.452677109 and -2.452677109.

Subproblem 1

x + -0.875 = 2.452677109 Simplifying x + -0.875 = 2.452677109 Reorder the terms: -0.875 + x = 2.452677109 Solving -0.875 + x = 2.452677109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = 2.452677109 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 2.452677109 + 0.875 x = 2.452677109 + 0.875 Combine like terms: 2.452677109 + 0.875 = 3.327677109 x = 3.327677109 Simplifying x = 3.327677109

Subproblem 2

x + -0.875 = -2.452677109 Simplifying x + -0.875 = -2.452677109 Reorder the terms: -0.875 + x = -2.452677109 Solving -0.875 + x = -2.452677109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = -2.452677109 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -2.452677109 + 0.875 x = -2.452677109 + 0.875 Combine like terms: -2.452677109 + 0.875 = -1.577677109 x = -1.577677109 Simplifying x = -1.577677109

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.327677109, -1.577677109}

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