Find two binomials whose product is x^2 - 6x + 9 and explain how you decided on those two binomials.
Thank you! :-)
To find two binomials whose product is x^2 - 6x + 9, we need to factorize the given quadratic expression. Here’s how you can do it:
Step 1: Write down the expression x^2 - 6x + 9.
Step 2: Identify if there are any common factors among the terms. In this case, all three terms have a common factor of 3.
Step 3: Factor out the common factor, which is 3, from each term. We get 3(x^2/3 - 6x/3 + 9/3).
Simplifying this gives us: 3(x^2/3 - 2x + 3).
Step 4: Now, focus on the expression inside the parentheses: x^2/3 - 2x + 3. We want to find two binomials that multiply to give this expression.
Step 5: To factorize this trinomial, we need to find two numbers that multiply to give the constant term (3) and add up to give the coefficient of the middle term (-2).
In this case, the numbers are -1 and -2. (Using trial and error or the quadratic formula, -1 and -2 are found to be the correct numbers.)
Step 6: Rewrite the middle term (-2x) using -1x and -2x: x^2/3 - 1x - 2x + 3.
Step 7: Group the terms and factor by grouping: (x^2/3 - 1x) + (-2x + 3).
Step 8: In each grouped pair, factor out the common factor. This gives us: x(x/3 - 1) - 2(x - 3).
Step 9: Finally, rewrite the expression with the factored terms: x(x - 3) - 2(x - 3).
Now we have our two binomials: (x - 3) and (x - 2).
So, the two binomials whose product is x^2 - 6x + 9 are (x - 3) and (x - 2).
since the coefficient of the square term at the front is 1, I just have to concentrate on the last term.
What two numbers when multiplied will give me 9 and when added will give me -6.
mmmh?
it can only be -3 and -3
so
x^2 - 6x + 9
= (x-3)(x-3) or (x-3)^2