Question

Solve the equation by factoring, by finding square roots, or by using the quadratic formula. $$8 c^{2}-27 c-24=-6$$

Short answer

The solutions to the equation are \(c_1 \approx 3.536\) and \(c_2 \approx -0.786\).

Step-by-step solution

Rearrage the equation

First, rearrange the equation to have all terms on one side and set to zero. Giving:\[ 8c^{2} - 27c - 18 = 0\]

Apply the quadratic formula

Then apply the quadratic formula \(c = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) where \(a = 8\), \(b = -27\), and \(c = -18\):\[c = \frac{-(-27) \pm \sqrt{(-27)^{2} - 4*8*(-18)}}{2*8}\]

Simplify the expression

Now simplify the expression that is inside the square root and the fraction in order to get the solution for c:\[c = \frac{27 \pm \sqrt{729 + 576}}{16} = \frac{27 \pm \sqrt{1305}}{16}\]

Calculate the values

Lastly, perform the operation to find the possible values for c, resulting in two solutions:\[c_1 = \frac{27 + \sqrt{1305}}{16} \approx 3.536\]\[c_2 = \frac{27 - \sqrt{1305}}{16} \approx -0.786\]