Let `f(x)=x^(2)+2x-5` and `g(x)=5x+30`
What are the roots of the equation g [f(x)]=0

Options

A. 1, -1
B. `-1,-1`
C. 1, 1
D. 0,1

Answer

To find the roots of the equation \( g(f(x)) = 0 \), we will follow these steps:

### Step 1: Define the functions
We have two functions:
- \( f(x) = x^2 + 2x - 5 \)
- \( g(x) = 5x + 30 \)

### Step 2: Substitute \( f(x) \) into \( g(x) \)
We need to find \( g(f(x)) \):
\[ g(f(x)) = g(x^2 + 2x - 5) \]
Substituting \( f(x) \) into \( g(x) \):
\[ g(f(x)) = 5(x^2 + 2x - 5) + 30 \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ g(f(x)) = 5(x^2 + 2x - 5) + 30 = 5x^2 + 10x - 25 + 30 \]
Combining like terms:
\[ g(f(x)) = 5x^2 + 10x + 5 \]

### Step 4: Set the equation to zero
Now we set \( g(f(x)) \) equal to zero:
\[ 5x^2 + 10x + 5 = 0 \]

### Step 5: Factor out the common term
We can factor out the common term 5:
\[ 5(x^2 + 2x + 1) = 0 \]
Dividing both sides by 5 (since 5 is not zero):
\[ x^2 + 2x + 1 = 0 \]

### Step 6: Factor the quadratic equation
The quadratic can be factored as:
\[ (x + 1)^2 = 0 \]

### Step 7: Solve for \( x \)
Setting the factor equal to zero gives:
\[ x + 1 = 0 \]
Thus, we find:
\[ x = -1 \]

### Conclusion
The root of the equation \( g(f(x)) = 0 \) is:
\[ x = -1 \]