Understanding Perfect Square Trinomials: Definition, Examples, and Factoring Process

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. This means the trinomial takes the form:

ax^2 + bx + c = (px + q)^2

where a, b, and c are constants, and p and q are coefficients of the binomial.

To determine if a trinomial is a perfect square, we can compare it with the general form of a perfect square trinomial. We need to check if all the terms in the original trinomial match those in the factored form.

Let’s look at an example to illustrate this concept:

Example: Determine if the trinomial 4x^2 + 4x + 1 is a perfect square.

Step 1: Compare the trinomial with the general form of a perfect square trinomial.

The general form of a perfect square trinomial is (px + q)^2, where p and q are coefficients of the binomial.

Comparing the trinomial 4x^2 + 4x + 1 with (px + q)^2, we see that:

a = 4 (coefficient of x^2)
b = 4 (coefficient of x)
c = 1 (constant term)

We need to determine the values of p and q to check if the trinomial is a perfect square.

Step 2: Determine the values of p and q.

To find the values of p and q, we can expand the expression (px + q)^2 and compare it with the given trinomial.

(px + q)^2 = p^2x^2 + 2pqx + q^2

Comparing the expanded form with the trinomial 4x^2 + 4x + 1, we get:

p^2 = 4 (a = 4)
2pq = 4 (b = 4)
q^2 = 1 (c = 1)

From the first equation (p^2 = 4), we find p = 2 or p = -2.
From the third equation (q^2 = 1), we find q = 1 or q = -1.

Step 3: Determine if the trinomial is a perfect square.

Now that we have the possible values of p and q, we substitute them back into the factored form (px + q)^2 and check if it matches the trinomial.

Possible factored forms are:
(2x + 1)^2 = 4x^2 + 4x + 1 (Matches the original trinomial)
(-2x + 1)^2 = 4x^2 – 4x + 1 (Does not match the original trinomial)

The original trinomial can be factored into (2x + 1)^2, so it is a perfect square trinomial.

In summary, a perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. To determine if a trinomial is a perfect square, we compare it with the general form of a perfect square trinomial and find the values of p and q. Then, we substitute these values back into the factored form and check if it matches the original trinomial.

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