The equation $x^2 + (m-3)x + m = 0$ has two real and distinct roots - Scottish Highers Maths - Question 10 - 2018

Expanding the discriminant yields:

$(m - 3)^2 - 4m = m^2 - 6m + 9 - 4m = m^2 - 10m + 9$

For the equation to have two real and distinct roots, the discriminant must be greater than zero:

$m^2 - 10m + 9 > 0$

We can factor this expression:

$(m - 1)(m - 9) > 0$

Now, we analyze when the product $(m - 1)(m - 9)$ is positive. The critical points are $m = 1$ and $m = 9$. Using a sign chart:

• For $m < 1$, both factors are negative (product is positive).
• For $1 < m < 9$, one factor is negative and one is positive (product is negative).
• For $m > 9$, both factors are positive (product is positive).

Thus, the intervals where the discriminant is positive are:

$m < 1 ext{ or } m > 9$

In conclusion, the range of values for $m$ for which the quadratic equation has two real and distinct roots is:

$m < 1 ext{ or } m > 9$