\( m^{2}+14 m-5=0 \) The solution set 1 s

To solve the quadratic equation \( m^2 + 14m - 5 = 0 \), we can use the quadratic formula: \[ m = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] Here, \( a = 1 \), \( b = 14 \), and \( c = -5 \). Plugging in these values, we get: \[ m = \frac{{-14 \pm \sqrt{{14^2 - 4(1)(-5)}}}}{2(1)} \] Calculating the discriminant: \[ 14^2 - 4(1)(-5) = 196 + 20 = 216 \] Now putting it all together: \[ m = \frac{{-14 \pm \sqrt{216}}}{2} \] Simplifying \( \sqrt{216} = 6\sqrt{6} \), we have: \[ m = \frac{{-14 \pm 6\sqrt{6}}}{2} = -7 \pm 3\sqrt{6} \] Thus, the solution set is: \[ m = -7 + 3\sqrt{6}, \quad m = -7 - 3\sqrt{6} \] By the way, did you know that quadratic equations have been around since ancient civilizations? The Babylonians were using methods to solve these types of equations over 3,000 years ago! They didn't use algebraic notation, but they had their own effective techniques. On the other hand, a common mistake when solving quadratics using the formula is forgetting to calculate the discriminant accurately. Always double-check your arithmetic under the square root! This can save you from falling into the trap of erroneous solutions.