Let's solve the given equation step-by-step:
[tex]\[ -b + \sqrt{60 - 6b} = -10 \][/tex]
### Step 1: Isolate the square root term
First, we want to isolate the square root term on one side of the equation. To do this, add [tex]\(b\)[/tex] to both sides:
[tex]\[ \sqrt{60 - 6b} = b - 10 \][/tex]
### Step 2: Square both sides
To eliminate the square root, square both sides of the equation:
[tex]\[ (\sqrt{60 - 6b})^2 = (b - 10)^2 \][/tex]
This simplifies to:
[tex]\[ 60 - 6b = (b - 10)^2 \][/tex]
### Step 3: Expand the squared term
Expand the right-hand side of the equation:
[tex]\[ 60 - 6b = b^2 - 20b + 100 \][/tex]
### Step 4: Rearrange into a standard quadratic equation
Bring all terms to one side to set the equation to 0:
[tex]\[ b^2 - 20b + 100 + 6b - 60 = 0 \][/tex]
Combine like terms:
[tex]\[ b^2 - 14b + 40 = 0 \][/tex]
### Step 5: Solve the quadratic equation
We can solve the quadratic equation using the quadratic formula [tex]\( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \)[/tex], where [tex]\( A = 1 \)[/tex], [tex]\( B = -14 \)[/tex], and [tex]\( C = 40 \)[/tex]:
Substitute these values into the formula:
[tex]\[ b = \frac{-(-14) \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 40}}{2 \cdot 1} \][/tex]
Simplify inside the square root:
[tex]\[ b = \frac{14 \pm \sqrt{196 - 160}}{2} \][/tex]
[tex]\[ b = \frac{14 \pm \sqrt{36}}{2} \][/tex]
[tex]\[ b = \frac{14 \pm 6}{2} \][/tex]
### Step 6: Find the two potential solutions
Calculate the two possible solutions:
[tex]\[ b = \frac{14 + 6}{2} = \frac{20}{2} = 10 \][/tex]
[tex]\[ b = \frac{14 - 6}{2} = \frac{8}{2} = 4 \][/tex]
### Step 7: Validate the solutions
To ensure the solutions are valid, substitute them back into the original equation:
1. For [tex]\( b = 10 \)[/tex]:
[tex]\[ -10 + \sqrt{60 - 6 \cdot 10} = -10 \][/tex]
[tex]\[ -10 + \sqrt{60 - 60} = -10 \][/tex]
[tex]\[ -10 + \sqrt{0} = -10 \][/tex]
[tex]\[ -10 = -10 \][/tex] (Valid)
2. For [tex]\( b = 4 \)[/tex]:
[tex]\[ -4 + \sqrt{60 - 6 \cdot 4} = -10 \][/tex]
[tex]\[ -4 + \sqrt{60 - 24} = -10 \][/tex]
[tex]\[ -4 + \sqrt{36} = -10 \][/tex]
[tex]\[ -4 + 6 = -10 \][/tex]
[tex]\[ 2 \neq -10 \][/tex] (Invalid)
### Final Answer
Given that the valid solution is [tex]\( b = 10 \)[/tex], the solution to the equation [tex]\( -b + \sqrt{60 - 6b} = -10 \)[/tex] is:
[tex]\[ b = 10 \][/tex]
