To solve the equation [tex]\( x^2 + 7x + 15 = -x - 1 \)[/tex] and find the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] of the quadratic equation, we need to first bring all terms to one side to set it equal to zero.
### Step-by-Step Solution:
1. Start with the given quadratic equation:
[tex]\[
x^2 + 7x + 15 = -x - 1
\][/tex]
2. Move all terms to one side to obtain a standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
x^2 + 7x + 15 + x + 1 = 0
\][/tex]
3. Combine like terms:
- Combine the [tex]\( x \)[/tex] terms: [tex]\( 7x + x = 8x \)[/tex]
- Combine the constant terms: [tex]\( 15 + 1 = 16 \)[/tex]
4. Write the simplified quadratic equation:
[tex]\[
x^2 + 8x + 16 = 0
\][/tex]
5. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the quadratic equation [tex]\( x^2 + 8x + 16 = 0 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex]) is [tex]\( 1 \)[/tex]
- The coefficient [tex]\( b \)[/tex] (the coefficient of [tex]\( x \)[/tex]) is [tex]\( 8 \)[/tex]
- The constant term [tex]\( c \)[/tex] is [tex]\( 16 \)[/tex]
Therefore, the solutions are:
[tex]\[
a = 1, \quad b = 8, \quad c = 16
\][/tex]
