Alright, let's solve the given quadratic equation using the quadratic formula step-by-step.
We have the equation in the form:
[tex]\[x = \frac{-8 \pm \sqrt{112}}{6}.\][/tex]
Step 1: Identify the components
- The term under the square root is called the discriminant, which is 112 in this case.
- The numerator before the square root is -8.
- The denominator is 6.
Step 2: Calculate the square root of the discriminant
- We need to find the value of \(\sqrt{112}\).
Step 3: Split the equation into two possible solutions
1. \(x_1 = \frac{-8 + \sqrt{112}}{6}\)
2. \(x_2 = \frac{-8 - \sqrt{112}}{6}\)
First, find \(\sqrt{112}\):
[tex]\[\sqrt{112} \approx 10.583005244258363.\][/tex]
Step 4: Calculate the two possible values of \(x\)
For \(x_1\):
[tex]\[x_1 = \frac{-8 + 10.583005244258363}{6}.\][/tex]
[tex]\[-8 + 10.583005244258363 = 2.583005244258363.\][/tex]
[tex]\[x_1 = \frac{2.583005244258363}{6} \approx 0.4305008740430605.\][/tex]
For \(x_2\):
[tex]\[x_2 = \frac{-8 - 10.583005244258363}{6}.\][/tex]
[tex]\[-8 - 10.583005244258363 = -18.583005244258363.\][/tex]
[tex]\[x_2 = \frac{-18.583005244258363}{6} \approx -3.097167540709727.\][/tex]
Therefore, the solutions to the equation are:
[tex]\[x_1 \approx 0.4305008740430605 \][/tex]
[tex]\[x_2 \approx -3.097167540709727.\][/tex]
