# Find the value of k if f(x) has three distinct real roots ?

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#x^3-3x+k=0#

##### 2 Answers

See explanation below

#### Explanation:

If equation has three real roots all of them are distinct, then

Developing:

We know that two polynomial expresions are equal if and only if his coefficients are equal. Then (Cardano formulae)

Now, if we take a look to

Adding +k we are traslating graph up or down if k is positive or negative respectively and this fact we can eliminate 2 real roots adding 2 complex roots.(if a polinomial has a complex root, then his conjugate is also root). Hope this helps

#### Explanation:

Let's start by taking a look at

graph{x^3-3x [-2 2 -3 3]}

The minimum and maximum are found by posing

With corresponding values

Adding

If we translate the function up three units, for example, the minimum would be above the

graph{x^3-3x+3 [-5 5 -1 6]}

Similarly, we can't translate the graph down more than two units: see this example with

graph{x^3-3x-3 [-5 5 -6 1]}

While any

graph{x^3-3x+1 [-3 3 -2 4]}