Perfect Square Trinomial
Recognize and factor perfect square trinomials in Grade 10 algebra: a²+2ab+b²=(a+b)² and a²-2ab+b²=(a-b)², and use this pattern to complete the square.
Key Concepts
Property: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 2ab + b^2 = (a b)^2$. A trinomial is a perfect square if its first and last terms are perfect squares, and the middle term is exactly twice the product of their square roots. Recognizing this pattern is a major shortcut.
Factor $x^2 14x + 49$: $(x 7)^2$ Factor $9m^2 + 24mn + 16n^2$: $(3m + 4n)^2$ Factor $y^2 + 20y + 100$: $(y + 10)^2$.
Spotting a perfect square trinomial is like finding a secret passage in a video game. If you see two perfect squares at the ends ($a^2$ and $b^2$) and the middle term is just $2ab$ (or $ 2ab$), you can skip all the hard work and jump straight to the answer: $(a+b)^2$ or $(a b)^2$.
Common Questions
How do you identify a perfect square trinomial?
A trinomial a²+2ab+b² or a²-2ab+b² is a perfect square. Check if the first and last terms are perfect squares and the middle term is ±2 times the product of their square roots.
How do you factor x²+12x+36?
Check: √(x²)=x, √36=6, and 2(x)(6)=12x ✓. So x²+12x+36 = (x+6)².
How are perfect square trinomials used in completing the square?
To complete the square for x²+bx, add (b/2)² to create the perfect square trinomial x²+bx+(b/2)² = (x+b/2)². This technique converts quadratics to vertex form.