Quadratics
Factoring to turn to Factored Form
Perfect Square Trinomials
Whenever a binomial or an expression is multiplied by itself, the resulting trinomial is a perfect square trinomial. These trinomials are the same as any other types of quadratic equations previously mentioned, except that when they are factored, the resulting factors are 2 same binomials.
The expression for a perfect square trinomial is a^2 + 2ab + b^2 which gets factored to (a + b)^2.
In order to factor a perfect square trinomial, find the values of a and b and then sub them both into the factored form expression (a + b)^2.
EXAMPLE
4v^2 + 20v + 25
This equation is in the form a^2 + 2ab + b^2, which will be factored to (a + b)^2. Therefore, we need to find our a and b values.
Now that we have our a and b values, we could sub in these values to the factored form (a + b)^2 to get the factored form of the expression.
Factored Form:
(2v + 5)^2
To check your answer sub in the values of a and b that you got into the perfect square trinomial expression and check if your answer is the same as the equation in the question.
a^2 + 2ab + b^2
(2v)^2 + 2(2v)(5) + (5)^2
4v^2 + 20v + 25
This may seem very simple, however, there is another model for perfect square trinomials in case of the term 2ab being negative.
This is represented as a^2 - 2ab + b^2, which gets factored to (a - b)^2.
EXAMPLE
25x^2 - 70x + 49
This equation is in the form a^2 - 2ab + b^2, which will be factored to (a - b)^2. Therefore, we need to find our a and b values.
Now that we have our a and b values, we could sub in these values to the factored form (a - b)^2 to get the factored form of the expression.
To check your answer sub in the values of a and b that you got into the perfect square trinomial expression and check if your answer is the same as the equation in the question.
a^2 - 2ab + b^2
(5x)^2 - 2(5x)(7) + (7)^2
25x^2 - 70x + 49
Factored Form:
(5x - 7)^2
Difference of Squares
Difference of squares is when a number that is being squared is subtracted from another number that is squred. This could be modelled as:
a^2 - b^2
The factored form of a^2 - b^2 is (a - b)(a + b).
For there to be a perfect square, the numbers that are being squared must be subtracted, if they are added, it is not a perfect square.
How to factor difference of squares?
EXAMPLE
100x^2 - 81
What we need to do is write both terms/numbers as squared.
100x^2 would be written a (10x)^2 and 81 would be written as 9^2.
Now the expression should look like:
(10x)^2 - (9)^2
Now this expression is in the form:
(a)^2 - (b)^2
10x is the a value and 9 is the b value.
Since the factored form of a difference of square is in the form (a - b)(a + b), sub in 'a' and 'b' into the expression and the final answer should be:
(10x - 9)(10x + 9)
To check your work, expand and simplify your answer to see if the answer is the same as the equation in the question.
(10x - 9) (10x + 9)
100x^2 -81