Answer :
To determine how the signs of [tex]\(b\)[/tex] and [tex]\(c\)[/tex] in the standard form [tex]\(x^2 + bx + c\)[/tex] affect the signs of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] in the factored form [tex]\((x + p)(x + q)\)[/tex], we need to delve into how the coefficients are related.
### Standard Form to Factored Form
The standard form of a quadratic equation is given by:
[tex]\[ x^2 + b x + c \][/tex]
The factored form of this quadratic equation can be written as:
[tex]\[ (x + p)(x + q) \][/tex]
When expanded, this becomes:
[tex]\[ x^2 + (p + q)x + pq \][/tex]
By comparing the coefficients from both forms, we get the following relationships:
1. [tex]\( b = p + q \)[/tex]
2. [tex]\( c = pq \)[/tex]
### Relationships Between [tex]\(b\)[/tex] and [tex]\(c\)[/tex] and the Signs of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]
1. When [tex]\( b \)[/tex] is positive and [tex]\( c \)[/tex] is positive:
- For the product [tex]\( pq \)[/tex] to be positive, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have the same sign.
- Since [tex]\( p + q = b \)[/tex] (a positive sum), both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be negative (as the sum of two negative numbers can be positive).
2. When [tex]\( b \)[/tex] is negative and [tex]\( c \)[/tex] is positive:
- Again, for [tex]\( pq \)[/tex] to be positive, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have the same sign.
- Since [tex]\( p + q = b \)[/tex] (a negative sum), both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be positive.
3. When [tex]\( b \)[/tex] is negative and [tex]\( c \)[/tex] is negative:
- For [tex]\( pq \)[/tex] to be negative, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have opposite signs.
- Since [tex]\( p + q = b \)[/tex] (a negative sum), one value must be more negative than the other is positive.
4. When [tex]\( b \)[/tex] is positive and [tex]\( c \)[/tex] is negative:
- For [tex]\( pq \)[/tex] to be negative, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have opposite signs.
- Since [tex]\( p + q = b \)[/tex] (a positive sum), one value must be more positive than the other is negative.
### Example
To illustrate a concrete example, consider [tex]\( b = 2 \)[/tex] and [tex]\( c = -3 \)[/tex]:
- These values suggest that [tex]\( b \)[/tex] is positive and [tex]\( c \)[/tex] is negative.
- Consequently, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have opposite signs.
Solving the quadratic equation [tex]\( x^2 + 2x - 3 = 0 \)[/tex], we get:
[tex]\[ (x - 1)(x + 3) = 0 \][/tex]
Thus, the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are:
[tex]\[ p = -3 \][/tex]
[tex]\[ q = 1 \][/tex]
These values satisfy:
- [tex]\( p + q = -3 + 1 = -2 \)[/tex], which is indeed [tex]\( b \)[/tex]
- [tex]\( pq = (-3) \times 1 = -3 \)[/tex], which is indeed [tex]\( c \)[/tex]
### Summary:
- [tex]\( b > 0 \)[/tex] and [tex]\( c > 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are both negative.
- [tex]\( b < 0 \)[/tex] and [tex]\( c > 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are both positive.
- [tex]\( b < 0 \)[/tex] and [tex]\( c < 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have opposite signs.
- [tex]\( b > 0 \)[/tex] and [tex]\( c < 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have opposite signs.
Each pair of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] solves the quadratic equation and maintains these relationships among the coefficients [tex]\( b \)[/tex] and [tex]\( c \)[/tex].
### Standard Form to Factored Form
The standard form of a quadratic equation is given by:
[tex]\[ x^2 + b x + c \][/tex]
The factored form of this quadratic equation can be written as:
[tex]\[ (x + p)(x + q) \][/tex]
When expanded, this becomes:
[tex]\[ x^2 + (p + q)x + pq \][/tex]
By comparing the coefficients from both forms, we get the following relationships:
1. [tex]\( b = p + q \)[/tex]
2. [tex]\( c = pq \)[/tex]
### Relationships Between [tex]\(b\)[/tex] and [tex]\(c\)[/tex] and the Signs of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]
1. When [tex]\( b \)[/tex] is positive and [tex]\( c \)[/tex] is positive:
- For the product [tex]\( pq \)[/tex] to be positive, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have the same sign.
- Since [tex]\( p + q = b \)[/tex] (a positive sum), both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be negative (as the sum of two negative numbers can be positive).
2. When [tex]\( b \)[/tex] is negative and [tex]\( c \)[/tex] is positive:
- Again, for [tex]\( pq \)[/tex] to be positive, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have the same sign.
- Since [tex]\( p + q = b \)[/tex] (a negative sum), both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be positive.
3. When [tex]\( b \)[/tex] is negative and [tex]\( c \)[/tex] is negative:
- For [tex]\( pq \)[/tex] to be negative, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have opposite signs.
- Since [tex]\( p + q = b \)[/tex] (a negative sum), one value must be more negative than the other is positive.
4. When [tex]\( b \)[/tex] is positive and [tex]\( c \)[/tex] is negative:
- For [tex]\( pq \)[/tex] to be negative, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have opposite signs.
- Since [tex]\( p + q = b \)[/tex] (a positive sum), one value must be more positive than the other is negative.
### Example
To illustrate a concrete example, consider [tex]\( b = 2 \)[/tex] and [tex]\( c = -3 \)[/tex]:
- These values suggest that [tex]\( b \)[/tex] is positive and [tex]\( c \)[/tex] is negative.
- Consequently, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have opposite signs.
Solving the quadratic equation [tex]\( x^2 + 2x - 3 = 0 \)[/tex], we get:
[tex]\[ (x - 1)(x + 3) = 0 \][/tex]
Thus, the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are:
[tex]\[ p = -3 \][/tex]
[tex]\[ q = 1 \][/tex]
These values satisfy:
- [tex]\( p + q = -3 + 1 = -2 \)[/tex], which is indeed [tex]\( b \)[/tex]
- [tex]\( pq = (-3) \times 1 = -3 \)[/tex], which is indeed [tex]\( c \)[/tex]
### Summary:
- [tex]\( b > 0 \)[/tex] and [tex]\( c > 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are both negative.
- [tex]\( b < 0 \)[/tex] and [tex]\( c > 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are both positive.
- [tex]\( b < 0 \)[/tex] and [tex]\( c < 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have opposite signs.
- [tex]\( b > 0 \)[/tex] and [tex]\( c < 0 \)[/tex]: [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have opposite signs.
Each pair of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] solves the quadratic equation and maintains these relationships among the coefficients [tex]\( b \)[/tex] and [tex]\( c \)[/tex].
