Answer :
To solve for the value of \( p \) in the quadratic equation \( px^2 - x + 1 = 0 \), given that the sum of its zeroes (or roots) is -2, follow these steps:
1. Identify the properties of a quadratic equation:
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of its roots \( (\alpha + \beta) \) is given by the formula:
[tex]\[ \alpha + \beta = -\frac{b}{a} \][/tex]
2. Substitute the coefficients:
In our given quadratic equation \( px^2 - x + 1 = 0 \), it's clear that:
[tex]\[ a = p, \quad b = -1, \quad \text{and} \quad c = 1 \][/tex]
3. Apply the sum of the roots formula:
According to the formula for the sum of the roots:
[tex]\[ \alpha + \beta = -\frac{b}{a} \][/tex]
Substituting the values of \( b \) and \( a \):
[tex]\[ \alpha + \beta = -\frac{-1}{p} = \frac{1}{p} \][/tex]
4. Use the given sum of the zeroes:
We are given that the sum of the zeroes is -2:
[tex]\[ \frac{1}{p} = -2 \][/tex]
5. Solve for \( p \):
To isolate \( p \), we solve the equation:
[tex]\[ \frac{1}{p} = -2 \implies p = -\frac{1}{2} \][/tex]
Hence, the value of \( p \) is:
[tex]\[ p = -0.5 \][/tex]
1. Identify the properties of a quadratic equation:
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of its roots \( (\alpha + \beta) \) is given by the formula:
[tex]\[ \alpha + \beta = -\frac{b}{a} \][/tex]
2. Substitute the coefficients:
In our given quadratic equation \( px^2 - x + 1 = 0 \), it's clear that:
[tex]\[ a = p, \quad b = -1, \quad \text{and} \quad c = 1 \][/tex]
3. Apply the sum of the roots formula:
According to the formula for the sum of the roots:
[tex]\[ \alpha + \beta = -\frac{b}{a} \][/tex]
Substituting the values of \( b \) and \( a \):
[tex]\[ \alpha + \beta = -\frac{-1}{p} = \frac{1}{p} \][/tex]
4. Use the given sum of the zeroes:
We are given that the sum of the zeroes is -2:
[tex]\[ \frac{1}{p} = -2 \][/tex]
5. Solve for \( p \):
To isolate \( p \), we solve the equation:
[tex]\[ \frac{1}{p} = -2 \implies p = -\frac{1}{2} \][/tex]
Hence, the value of \( p \) is:
[tex]\[ p = -0.5 \][/tex]
