Question

consider the function ( g(x)=-x^{2}+14x + 39 ).
( g(x) ) has a discriminant of choose your answer... this means that ( g(x) ) will have choose your answer... real roots.
( g(x) ) has a choose your answer... when ( x = 7 ) since choose your answer...

Explanation:

Step1: Calculate the discriminant

For a quadratic function \(y = ax^{2}+bx + c\), the discriminant is \(\Delta=b^{2}-4ac\).
In the function \(g(x)=-x^{2}+14x + 39\), \(a=-1\), \(b = 14\), \(c = 39\).
So \(\Delta=(14)^{2}-4\times(-1)\times39\)
\(=196 + 156\)
\(=352\)

Step2: Determine the number of real roots

Since \(\Delta=352>0\), the quadratic function \(g(x)\) has two distinct real roots.

Step3: Analyze the vertex

The \(x\) - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is \(x=-\frac{b}{2a}\).
Here \(x =-\frac{14}{2\times(-1)}=7\).
For \(y=-x^{2}+14x + 39\), \(a=-1<0\), so the parabola opens downwards.
When \(x = 7\), \(g(x)\) has a maximum value.

Answer:

• Discriminant: \(352\)
• Number of real roots: two
• \(g(x)\) has a maximum when \(x = 7\) since the coefficient of \(x^{2}\) (\(a=-1\)) is negative.