If x² + mx + m is a perfect square trinomial, which equation must be true?

For a quadratic expression to be a perfect square trinomial, it must take the form of (x + a)², which expands to x² + 2ax + a².

If we compare this with the expression x² + mx + m, we can see that:

• The coefficient of x (which is m) should equal 2a.
• The constant term (which is m) should equal a².

From these relationships, we can derive two equations:

1. m = 2a
2. m = a²

Now, substituting the first equation into the second:

m = (m/2)²

Multiplying both sides by 4 to eliminate the fraction gives:

4m = m²

Rearranging this leads to:

m² - 4m = 0

This can be factored as:

m(m - 4) = 0

Thus, the values for m that make the expression a perfect square trinomial are:

• m = 0
• m = 4

In conclusion, if the expression x² + mx + m is a perfect square trinomial, then the equation m(m – 4) = 0 must be true, leading to the possible values for m of 0 or 4.