Question

Solve the equation. Tell which solution method you used. \(27+6 w-w^{2}=0\)

Short answer

The solutions are \(w=9\) and \(w=-3\). The quadratic formula was used to find the solutions.

Step-by-step solution

Identify the coefficients

From the given equation \(27+6 w-w^{2}=0\), rearranged to standard form is \(-w^{2}+6w+27=0\). The coefficients identified are \(a=-1\), \(b=6\) and \(c=27\).

Calculate the discriminant

The discriminant is calculated using the formula \(\Delta = b^{2}-4ac = (6)^{2}-4*(-1)*27 = 36 +108 = 144\). As \(\Delta\) is greater than 0, the roots are real and distinct.

Use the quadratic formula

The quadratic formula is \(w = \frac{{-b \pm \sqrt{{\Delta}}}}{2a}\). By substituting \(a=-1\), \(b=6\) and \(\Delta=144\), we get \(w1 = \frac{{-6-12}}{-2} = 9\) and \(w2 = \frac{{-6+12}}{-2} = -3\).