# What is the value of c such that: x^2 + 14x + c, is a perfect-square trinomial?

##### 1 Answer

Consider the quadratic equation

Two identical solutions! Recall that the solutions of a quadratic equation are the x intercepts on the corresponding quadratic function.

So, the solutions to the equation

Similarly, the solutions to the equation

Since there is really only one solution to

Now, think of the discriminant of a quadratic equation. If you don't have previous experience with it, don't fret.

We use the discriminant,

When the discriminant equals less than **no solution**. When the discriminant equals exactly zero, the equation will have exactly **one solution**. When the discriminant equals any number more than zero, there will be exactly **two solutions**. If the number in question that you get as a result is a perfect square in the latter case, the equation will have two rational solutions. If not, it will have two irrational solutions.

I've already shown that when you have a perfect square trinomial, you will have two identical solutions, which is equal to one solution. Hence, we can set the discriminant to

Where

Thus, the perfect square trinomial with

**Practice exercises:**

- Using the discriminant, determine the values of
#a, b, or c# that render the trinomials perfect squares.

a)

b)

c)

Hopefully this helps, and good luck!