91Ó°ÊÓ

The discriminant is a key concept in understanding the nature of a quadratic function's roots. In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated using the formula: \( \Delta = b^2 - 4ac \).

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, meaning the graph touches the x-axis at one point.
• If \( \Delta < 0 \), there are no real roots, and the graph does not intersect the x-axis at all.

In our exercise, the quadratic function is \( g: x \mapsto mx^2 + 6x + m \). Applying the discriminant formula involves identifying \( a = m \), \( b = 6 \), and \( c = m \).
Once the discriminant is expressed as \( 36 - 4m^2 \), setting \( \Delta < 0 \) ensures that the graph does not touch the x-axis, confirming no real roots exist.

To find values of \( m \) where the graph does not intersect the x-axis, we need to solve the inequality derived from the discriminant. After calculating the discriminant as \( 36 - 4m^2 \), the inequality we want to solve is \( 36 - 4m^2 < 0 \).
By rearranging, we have \( 36 < 4m^2 \), which simplifies to \( 9 < m^2 \) by dividing the entire inequality by 4.
Taking the square root of both sides, we attain the condition \( |m| > 3 \). This can be interpreted as two cases due to the absolute value:

• \( m < -3 \)
• \( m > 3 \)

This means for the graph to not touch the x-axis, \( m \) must be less than -3 or greater than 3. With these conditions, the inequality ensures the graph of the function never touches the x-axis.

Real roots are the solutions where a quadratic function intersects the x-axis. For any quadratic equation \( ax^2 + bx + c = 0 \), the type and number of real roots can be determined using the discriminant.
The discriminant helps in predicting the nature of the roots:

• Two real roots occur if the discriminant \( \Delta > 0 \).
• A single real root, or a repeated root, occurs if \( \Delta = 0 \).
• No real roots arise if \( \Delta < 0 \), which means the graph does not touch or cross the x-axis.

In our function, \( g: x \mapsto mx^2 + 6x + m \), setting \( \Delta < 0 \) showed the absence of real roots. It resulted in the solution \( m < -3 \) or \( m > 3 \) to ensure the function graph does not touch the x-axis, solidifying the understanding of when a function will not have any real roots.