Certainly! Let's solve the quadratic equation step-by-step.
### Step 1: Rearrange the equation \(\ -b^2-18b-8=-6b^2\)
Move all terms to one side of the equation to set it to 0:
[tex]\[ -b^2 - 18b - 8 \ + \ 6b^2 = 0 \][/tex]
### Step 2: Combine like terms
[tex]\[ -b^2 \ + \ 6b^2 - 18b - 8 = 0 \][/tex]
[tex]\[ 5b^2 - 18b - 8 = 0 \][/tex]
Now, we have a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 5 \), \( b = -18 \), and \( c = -8 \).
### Step 3: Calculate the discriminant
The discriminant \(\Delta\) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values of \(a\), \(b\), and \(c\):
[tex]\[ \Delta = (-18)^2 - 4 \cdot 5 \cdot (-8) \][/tex]
[tex]\[ \Delta = 324 + 160 \][/tex]
[tex]\[ \Delta = 484 \][/tex]
### Step 4: Solve using the quadratic formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Using the values \(a = 5\), \(b = -18\), and \(\Delta = 484\):
Calculate the first root (\(x_1\)):
[tex]\[ x_1 = \frac{-(-18) + \sqrt{484}}{2 \cdot 5} \][/tex]
[tex]\[ x_1 = \frac{18 + 22}{10} \][/tex]
[tex]\[ x_1 = \frac{40}{10} \][/tex]
[tex]\[ x_1 = 4.0 \][/tex]
Calculate the second root (\(x_2\)):
[tex]\[ x_2 = \frac{-(-18) - \sqrt{484}}{2 \cdot 5} \][/tex]
[tex]\[ x_2 = \frac{18 - 22}{10} \][/tex]
[tex]\[ x_2 = \frac{-4}{10} \][/tex]
[tex]\[ x_2 = -0.4 \][/tex]
### Step 5: The solutions
The two rational solutions to the quadratic equation \( 5b^2 - 18b - 8 = 0 \) are:
[tex]\[ b = 4.0 \quad \text{and} \quad b = -0.4 \][/tex]
Therefore, the solutions are:
[tex]\[ b = 4.0 \][/tex]
[tex]\[ b = -0.4 \][/tex]
