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If -1 and 2 are roots of the equation x^...

If -1 and 2 are roots of the equation `x^2+bx+c=0`, then c is equal to:

A

1

B

2

C

-1

D

-2

Text Solution

AI Generated Solution

The correct Answer is:
To find the value of \( c \) in the equation \( x^2 + bx + c = 0 \) given that the roots are \( -1 \) and \( 2 \), we can use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots. ### Step-by-Step Solution: 1. **Identify the roots**: The roots of the equation are given as \( r_1 = -1 \) and \( r_2 = 2 \). 2. **Use Vieta's Formulas**: According to Vieta's formulas: - The sum of the roots \( r_1 + r_2 = -\frac{b}{1} \) (since the coefficient of \( x^2 \) is 1). - The product of the roots \( r_1 \cdot r_2 = \frac{c}{1} \). 3. **Calculate the sum of the roots**: \[ r_1 + r_2 = -1 + 2 = 1 \] Therefore, we have: \[ 1 = -b \implies b = -1 \] 4. **Calculate the product of the roots**: \[ r_1 \cdot r_2 = -1 \cdot 2 = -2 \] Therefore, we have: \[ -2 = c \] 5. **Conclusion**: Thus, the value of \( c \) is \( -2 \). ### Final Answer: \[ c = -2 \]
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