If the equation x^2−4x + k =1 has no real roots, then the range of values of k is A. k >4 B. k >=4 C. k >5 D. k >=5
The quadratic equation `ax^2+bx+c = 0` has three kinds of roots.
1. Two distinct real roots
2. Complex roots or no real roots
3. Identical roots
The type of root for a quadratic equation is determine by the discriminant(Delta).
1. Two distinct real roots (`Delta>0` )
2. Complex roots or no real roots (`Delta<0` )
3. Identical roots (`Delta = 0` )
`Delta = b^2-4ac`
For our question we need the second type of roots where `Delta<0` .
`x^2-4x + k =1`
`Delta = (-4)^2-4xx1xxk = 4(4-k)`
So the answer is k>4. Correct answer is at option A.