Question

Solve the equation by completing the square. $$6 x^{2}+24 x-41=0$$

Short answer

The solutions are \( x = -2 + \sqrt{\frac{65}{6}}\) and \( x = -2 - \sqrt{\frac{65}{6}}\)

Step-by-step solution

Arrange the equation in a suitable format

The given equation is \(6x^2 + 24x - 41 = 0\). It is in the format \(ax^2 + bx + c = 0\), where \(a = 6\), \(b = 24\) and \(c = -41\). To make the calculation easier, it might be useful to factor out the value a from the square component, getting \(x^2 + 4x - \frac{41}{6} = 0\)

Completing the square

\ Completing the square involves adding and subtracting the square of half the coefficient of x inside the square. That gives \(x^2 + 4x + (4/2)^2 - (4/2)^2 - \frac{41}{6} = 0\). This simplifies to \( (x + 2)^2 - 4 - \frac{41}{6} = 0 \

Solving the equation

\ Now solve the equation step by step. First, combine like terms to get: \( (x + 2)^2 = \frac{41}{6} + 4 \) or \( (x + 2)^2 = \frac{65}{6}\). Next, find the square root on both sides, giving \( x + 2 = \pm \sqrt{\frac{65}{6}}\). Lastly, subtract 2 from both sides to solve for x, \( x = -2 \pm \sqrt{\frac{65}{6}}\)