Hi! I have a few questions about Factoring Quadratic Expressions.
My first one is 12x^2+x=6. How would you facor that?
My second is how would x^4-1 equal the answer 1(x^4-1)(x^2=1)(x^2-1)?
Finally, how would you factor something such as -3x(x+2)-8(x+2), when there is no common factor other than one?
Thank you so much for your assistance! :)
12 x^2 + x - 6 = 0
I solved the quadratic and got 2/3 and -3/4 so then tried
(3x+2)(4x-3)=0
x^4-1 is difference of two squares
(x^2-1)(x^2+1)
then
(x-1)(x+1)(x^2+1)
for 12x^2+x=6 write is as
12x^2+x-6 = 0
then (3x-2)(4x+3) = 0
x^4-1 factors to
(x^2 -1)(x^2 + 1)
=(x-1)(x+1)(x^2+1)
for -3x(x+2)-8(x+2) you should see that ((x+2) is a common factor, so
-3x(x+2)-8(x+2)
= (x+2)(-3x-8)
= -(x+2)(3x+8)
-3x(x+2)-8(x+2)
sure there is a common factor -- (x+2)
(x+2)(-3x-8)
(x+2)(-1)(3x+8)
Hello! I'll be happy to help you with your questions about factoring quadratic expressions.
1. To factor the expression 12x^2 + x = 6:
Step 1: Move all terms to one side to set the equation equal to zero:
12x^2 + x - 6 = 0
Step 2: Check if the quadratic expression can be factored further. In this case, it can be factored using the factoring method called "grouping."
Step 3: Look for two numbers that multiply to give the product of the coefficient of x^2 (12) and the constant term (-6), which is -72.
The numbers that meet this criteria are -8 and 9 because (-8) * (9) = -72 and (-8) + (9) = 1.
Step 4: Rewrite the middle term (x) using the numbers found from step 3:
12x^2 - 8x + 9x - 6 = 0
Step 5: Now, group the terms:
(12x^2 - 8x) + (9x - 6) = 0
Step 6: Factor out the greatest common factor from each group:
4x(3x - 2) + 3(3x - 2) = 0
Step 7: Notice that we now have a common factor of (3x - 2) in both groups, so factor it out:
(4x + 3)(3x - 2) = 0
And that's the factored form of the quadratic expression 12x^2 + x - 6.
2. To factor the expression x^4 - 1:
Step 1: Notice that x^4 - 1 can be written as (x^2)^2 - 1^2.
Step 2: Use the difference of squares formula, which states that a^2 - b^2 can be factored as (a - b)(a + b).
Applying this to our expression, we get:
(x^2 - 1)(x^2 + 1)
Step 3: The expression x^2 - 1 can be further factored as (x - 1)(x + 1).
So, the final factored form of x^4 - 1 is:
(x - 1)(x + 1)(x^2 + 1).
3. To factor the expression -3x(x + 2) - 8(x + 2):
Step 1: Notice that (x + 2) is a common factor in both terms.
Step 2: Factor out (x + 2) from each term:
(x + 2)(-3x - 8)
And that's the factored form of the expression -3x(x + 2) - 8(x + 2). Even if there are no other common factors besides one, we still factor out the common factor that is present in both terms.
I hope this helps! If you have any further questions, feel free to ask.