Answer :
Sure, let’s solve the problem step by step. We are given a quadratic equation [tex]\( p x^2 + q = 0 \)[/tex], and we know that the roots differ by 1. We need to find the relationship between [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
#### Step 1: Understanding the quadratic equation
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this case:
- Coefficient of [tex]\( x^2 \)[/tex] is [tex]\( p \)[/tex]
- Coefficient of [tex]\( x \)[/tex] is 0 (since there is no [tex]\( x \)[/tex]-term in the equation)
- Constant term is [tex]\( q \)[/tex]
#### Step 2: Using Vieta’s formulas
By Vieta’s formulas:
- The sum of the roots [tex]\( \alpha + \beta \)[/tex] is given by [tex]\( -\frac{b}{a} \)[/tex]. Since [tex]\( b = 0 \)[/tex] and [tex]\( a = p \)[/tex]:
[tex]\[ \alpha + \beta = -\frac{0}{p} = 0 \][/tex]
- The product of the roots [tex]\( \alpha \beta \)[/tex] is given by [tex]\( \frac{c}{a} \)[/tex]. Here [tex]\( c = q \)[/tex] and [tex]\( a = p \)[/tex]:
[tex]\[ \alpha \beta = \frac{q}{p} \][/tex]
#### Step 3: Roots difference condition
Given that the roots differ by 1, we can write:
[tex]\[ \alpha - \beta = 1 \quad \text{or} \quad \beta - \alpha = 1 \][/tex]
#### Step 4: Solving the system of equations
Using the conditions:
[tex]\[ \alpha + \beta = 0 \quad \text{and} \quad \alpha - \beta = 1 \][/tex]
We can solve these equations to find [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
1. Adding the two equations:
[tex]\[ (\alpha + \beta) + (\alpha - \beta) = 0 + 1 \implies 2\alpha = 1 \implies \alpha = \frac{1}{2} \][/tex]
2. Substituting [tex]\( \alpha = \frac{1}{2} \)[/tex] into [tex]\( \alpha + \beta = 0 \)[/tex]:
[tex]\[ \frac{1}{2} + \beta = 0 \implies \beta = -\frac{1}{2} \][/tex]
So, the roots are [tex]\( \alpha = \frac{1}{2} \)[/tex] and [tex]\( \beta = -\frac{1}{2} \)[/tex].
#### Step 5: Product of the roots
Using the product of the roots:
[tex]\[ \alpha \beta = \frac{q}{p} \][/tex]
Substitute [tex]\( \alpha = \frac{1}{2} \)[/tex] and [tex]\( \beta = -\frac{1}{2} \)[/tex]:
[tex]\[ \left( \frac{1}{2} \right) \left( -\frac{1}{2} \right) = \frac{q}{p} \][/tex]
[tex]\[ - \frac{1}{4} = \frac{q}{p} \][/tex]
#### Step 6: Relationship between [tex]\( p \)[/tex] and [tex]\( q \)[/tex]
We can rearrange the equation:
[tex]\[ p \cdot -\frac{1}{4} = q \][/tex]
[tex]\[ p = -4q \][/tex]
Thus, the relationship between [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is:
[tex]\[ p + 4q = 0 \][/tex]
The correct answer is:
[tex]\[ \boxed{p + 4q = 0} \][/tex]
#### Step 1: Understanding the quadratic equation
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this case:
- Coefficient of [tex]\( x^2 \)[/tex] is [tex]\( p \)[/tex]
- Coefficient of [tex]\( x \)[/tex] is 0 (since there is no [tex]\( x \)[/tex]-term in the equation)
- Constant term is [tex]\( q \)[/tex]
#### Step 2: Using Vieta’s formulas
By Vieta’s formulas:
- The sum of the roots [tex]\( \alpha + \beta \)[/tex] is given by [tex]\( -\frac{b}{a} \)[/tex]. Since [tex]\( b = 0 \)[/tex] and [tex]\( a = p \)[/tex]:
[tex]\[ \alpha + \beta = -\frac{0}{p} = 0 \][/tex]
- The product of the roots [tex]\( \alpha \beta \)[/tex] is given by [tex]\( \frac{c}{a} \)[/tex]. Here [tex]\( c = q \)[/tex] and [tex]\( a = p \)[/tex]:
[tex]\[ \alpha \beta = \frac{q}{p} \][/tex]
#### Step 3: Roots difference condition
Given that the roots differ by 1, we can write:
[tex]\[ \alpha - \beta = 1 \quad \text{or} \quad \beta - \alpha = 1 \][/tex]
#### Step 4: Solving the system of equations
Using the conditions:
[tex]\[ \alpha + \beta = 0 \quad \text{and} \quad \alpha - \beta = 1 \][/tex]
We can solve these equations to find [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
1. Adding the two equations:
[tex]\[ (\alpha + \beta) + (\alpha - \beta) = 0 + 1 \implies 2\alpha = 1 \implies \alpha = \frac{1}{2} \][/tex]
2. Substituting [tex]\( \alpha = \frac{1}{2} \)[/tex] into [tex]\( \alpha + \beta = 0 \)[/tex]:
[tex]\[ \frac{1}{2} + \beta = 0 \implies \beta = -\frac{1}{2} \][/tex]
So, the roots are [tex]\( \alpha = \frac{1}{2} \)[/tex] and [tex]\( \beta = -\frac{1}{2} \)[/tex].
#### Step 5: Product of the roots
Using the product of the roots:
[tex]\[ \alpha \beta = \frac{q}{p} \][/tex]
Substitute [tex]\( \alpha = \frac{1}{2} \)[/tex] and [tex]\( \beta = -\frac{1}{2} \)[/tex]:
[tex]\[ \left( \frac{1}{2} \right) \left( -\frac{1}{2} \right) = \frac{q}{p} \][/tex]
[tex]\[ - \frac{1}{4} = \frac{q}{p} \][/tex]
#### Step 6: Relationship between [tex]\( p \)[/tex] and [tex]\( q \)[/tex]
We can rearrange the equation:
[tex]\[ p \cdot -\frac{1}{4} = q \][/tex]
[tex]\[ p = -4q \][/tex]
Thus, the relationship between [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is:
[tex]\[ p + 4q = 0 \][/tex]
The correct answer is:
[tex]\[ \boxed{p + 4q = 0} \][/tex]
