Question

Suppose the functions f(x)= -x2 +4x-p and g(x)=x2-px+9 has no real roots for some P belong to integer. Find the value of P.​

Answer 1

Step-by-step explanation:

The value of pp is 5.

Step-by-step explanation:

Given the functions,

f(x)=-x^2+4x-pf(x)=−x

2

+4x−p

and

g(x)=x^2-px+9g(x)=x

2

−px+9

We need to find the value of pp so that these functions have no real roots.

For a quadratic equation of the form ax^2+bx+cax

2

+bx+c has no real roots if its discriminant b^2-4acb

2

−4ac is negative.

So we need to find the discriminant of the functions given.

\begin{gathered}discriminant \ of \ f(x) =4^2-(4\times (-1)\times (-p))=16-4p\\\\discriminant \ of \ g(x) =(-p)^2-4\times 1\times 9=p^2-36\end{gathered}

discriminant of f(x)=4

2

−(4×(−1)×(−p))=16−4p

discriminant of g(x)=(−p)

2

−4×1×9=p

2

−36

For the function f(x)f(x) , the discriminant is negative implies,

\begin{gathered}\implies 16-4p < 0\\\\\implies 4p > 16\\\\\implies p > \frac{16}{4}=4\end{gathered}

⟹16−4p<0

⟹4p>16

⟹p>

4

16

=4

Similarly, For the function g(x)g(x) , the discriminant is negative implies,

\begin{gathered}\implies p^2-36 < 0\\\\\implies p^2 < 36\\\\\implies p < 6 \ \ \ or\ \ \ p < -6\end{gathered}

⟹p

2

−36<0

⟹p

2

<36

⟹p<6 or p<−6

So from these observations, pp should be greater than 4 and less than 6. Since pp is an integer, the value of pp should be 5.