To determine the value of c that makes the expression x² + 14x + c a perfect square trinomial, we need to understand the structure of perfect square trinomials.
A perfect square trinomial takes the form (a + b)², which expands to a² + 2ab + b². In our case, we can compare this with the given expression.
The first term, x², shows that a = x. The middle term is 14x, which corresponds to 2ab. If we set b to be a value we need to find, we can set up the equation:
2ab = 14x
Substituting a = x:
2(x)b = 14x
If we simplify this, we get:
2b = 14
From this, we find that:
b = 7
Now, we have the value of b. The last term, c, in our expression corresponds to b²:
c = b² = 7² = 49
Thus, the value of c that makes x² + 14x + c a perfect square trinomial is 49.