If the equation *x^2*−4*x *+ *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

### 1 Answer | Add Yours

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)`

`Delta<0`

`4(4-k)<0`

`4<k`

*So the answer is k>4. Correct answer is at option A.*

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