Question

If one of the root of the quadratic equation kx2 + (a+b) x + ab = 0 is – b, find the value of k

Answer 1

The value of
`\( k \) is \( 1 \).`

Step-by-step explanation:

The quadratic equation given is
`\( kx^2 + (a+b)x + ab = 0 \),` and one of its roots is given as
`\( -b \). In a quadratic equation \( ax^2 + bx + c = 0 \),` the sum of the roots is
`\( (-b)/(a) \),` and the product of the roots is
`\( (c)/(a) \).` In this case, the sum of the roots is
`\( (-(a+b))/(k) \) and the product is \( (ab)/(k) \).` Since one of the roots is given as
`\( -b \), the other root is \( (ab)/(k) \).` The sum of the roots is therefore
`\( -b + (ab)/(k) \),` and according to the given information, this is equal to
`\( -b \).`

Setting up the equation:

`\[ -b + (ab)/(k) = -b \]`

Simplifying, we get
`\( (ab)/(k) = 0 \), which implies \( ab = 0 \)`. Since
`\( a \) and \( b \)`are the coefficients in the quadratic equation, at least one of them must be zero. Assuming
`\( b \) is non-zero, \( a \)` must be zero. Therefore,
`\( k \) must be \( 1 \)` to satisfy the given conditions.