Given the quadratic equation \( x^2 + ax - b = 0 \) and knowing \( a \) and \( b \) are the roots of this equation, we can use Vieta's formulas to find them. Vieta's formulas state that for a quadratic equation of the form \( x^2 + px + q = 0 \) with roots \( r \) and \( s \):
1. The sum of the roots \( r \) and \( s \) is given by \( -p \).
2. The product of the roots \( r \) and \( s \) is given by \( q \).
Let's apply this to our equation \( x^2 + ax - b = 0 \):
1. The sum of the roots \( (a + b) \) should be equal to \(-a\).
2. The product of the roots \( (a \cdot b) \) should be equal to \( -b \).
Given the options, let's analyze which pairs satisfy both conditions:
(a) \( a = -1 \) and \( b = 2 \)
- Sum of the roots: \( -1 + 2 = 1 \)
- Product of the roots: \( -1 \times 2 = -2 \)
(b) \( a = 1 \) and \( b = 2 \)
- Sum of the roots: \( 1 + 2 = 3 \)
- Product of the roots: \( 1 \times 2 = 2 \)
(c) \( a = -2 \) and \( b = 1 \)
- Sum of the roots: \( -2 + 1 = -1 \)
- Product of the roots: \( -2 \times 1 = -2 \)
(d) \( a = 2 \) and \( b = -1 \)
- Sum of the roots: \( 2 + (-1) = 1 \)
- Product of the roots: \( 2 \times (-1) = -2 \)
For choices \(a\), \(c\), and \(d\), re-evaluating:
(a)
- Sum: \( 1 \neq -(-1) \) -> Incorrect for sum
- Product: \( -2 == -2 \) -> Correct for product
(b)
- Sum: \( 3 \neq -1 \) -> Incorrect for sum
- Product: \( 2 == 2 \) -> Correct for product
(c)
- Sum: \( -1 == -(-2) \) -> Incorrect for sum
- Product: \( -2 == -2 \) -> Correct for product
(d)
- Sum: \( 1 == -2 \) -> correct for sum
- Product: \( -2 == -2 \) -> correct for product
Given all pairs correctly summing the correct a pair and products correct, the suitable group is the final selection:
(a = -1 , b= 2)
So the correct answer is the selection:
which is: (d)
