What is a Perfect Square
Identify and factor perfect square trinomials: recognize the pattern a^2+2ab+b^2=(a+b)^2 to factor quickly and apply it to completing the square and solving quadratic equations.
Key Concepts
Property A perfect square trinomial comes from squaring a binomial, like $(x + n)^2 = x^2 + 2nx + n^2$. The key relationship is that the constant term, $c$, is the square of half the coefficient of the $x$ term, $b$. In other words, the magic formula is $c = (\frac{b}{2})^2$.
In $x^2 + 8x + 16$, half of $b=8$ is $4$, and $4^2=16$. This is a perfect square: $(x+4)^2$. In $x^2 + 10x + 25$, half of $b=10$ is $5$, and $5^2=25$. This is a perfect square: $(x+5)^2$. In $x^2 12x + 36$, half of $b= 12$ is $ 6$, and $( 6)^2=36$. This is a perfect square: $(x 6)^2$.
Imagine building a literal square with algebra tiles. A perfect square trinomial is an equation that forms a perfect, gap free square. This special relationship between the $b$ and $c$ terms is the secret to making this happen, and it's the foundation for the 'completing the square' method you'll use later on.
Common Questions
What is a perfect square trinomial?
A perfect square trinomial is a trinomial that factors into the square of a binomial. The pattern is a^2+2ab+b^2=(a+b)^2 or a^2-2ab+b^2=(a-b)^2. Recognizing this pattern speeds up factoring and completing-the-square problems.
How do you check if a trinomial is a perfect square?
Check that the first and last terms are perfect squares, and that the middle term equals twice the product of their square roots. For x^2+6x+9: sqrt(x^2)=x, sqrt(9)=3, and 2(x)(3)=6x matches the middle term, confirming (x+3)^2.
How are perfect square trinomials used when completing the square?
Completing the square deliberately creates a perfect square trinomial by adding (b/2)^2 to both sides of x^2+bx=c. Once the left side matches the perfect square pattern, it factors as (x+b/2)^2, making the equation solvable with the square root property.