Can you explain to me how I would factorise 15a^2 - 14a - 8.?

Question

Can you explain to me how I would factorise 15a^2 - 14a - 8. Giving me explanation's of your method.

Answer 1

Start with the 15... it could be 1 x 15 or 3 x 5.

Now look at -8 ... it could be 1 x -8, -1 x 8, 2 x -4, -2 x -4

Just try these 8 combinations and see which gives you -14 for the middle coefficient.

(a + 1)(15a - 8) --> 7a
(a - 8)(15a + 1) --> -119a
(3a + 1)(5a - 8) --> -19a
(3a -8)(5a + 1) --> -37a
(a + 2)(15a - 4) --> 26a
(a - 4)(15a + 2) --> -58a
(3a + 2)(5a - 4) --> -2a
(3a -4)(5a + 2) --> -14a <-- Finally!

The answer is:
(3a - 4)(5a + 2) = 0

And the roots would be a = 4/3 or a = -2/5

However, when factoring gets this complicated, you might switch to completing the square or using the quadratic formula.

Using the quadratic formula for a = 15, b = -14, c = -8 you would get:

[-b +/- sqrt(b^2 - 4ac) ] / 2a

= [14 +/- sqrt((-14)^2 - 4(15)(-8)) ] / 30
= 7/15 +/- sqrt(196 + 480) / 30
= 7/15 +/- sqrt(676) / 30
= 7/15 +/- 26 / 30
= 7/15 +/- 13 / 15
= 20/15 or -6/15
= 4/3 or -2/5

Turning it back into equations:
(a - 4/3)(a + 2/5) = 0
(3a - 4)(5a + 2) = 0

Answer 2

The method is educated guessing.
1. 15 can only be factored as 15*1or 3*5. I am going to make my trials with 3*5 first and only go to 15*1 if I get failures.
2. 8 can be factored only as 1*8 and 4*2, so I have more trials here.
3. First guess at the set up is (3a - ?)(5a + ?). Because the middle 14a is negative, I will have one of each sign
4. If the 4 goes with the 3a, I have big -20a and a plus something to counterbalance it. If i have a small number in the second spot times the big number, I might end up lot large enough to make the right answer.
5. The partial is (3a-4)(5a+?). The pair to 4 is 2.
6. I test (3a - 4)(5a + 2) and get 15a^2 -20a+6a-8. This is the staring problem and so must be the answer.
7. You can run all the other possibles and they don't work.
(3a-2)(5a+4)= 15a^2+2a-8
(3a-8)(5a+1) = 15a^2-37a-8
(3a-1)(5a+8) = 15a^2-19a-8
(a-2)(15a+4) = 15a^2-26a-8
(a-4)(15a+2) = 15a^2-58a-8
(a-1)(15a+8) = 15a^2-7a-8
(a-8)(15a+1) = 15a^2+7a-8

Answer 3

1 must admit that solving the equation with the formula is the best option for this type of problem

solving 15a^2 - 14a - 8=0 with the formula gives

a=4/3 or -2/5

(a+2/5)(a-4/3)=0

{5(a+2/5)}{3(a-4/3)}=(5a+2)(3a-4)=0

hence, 15a^2 - 14a - 8 =(5a+2)(3a-4)

i hope that this helps

Answer 4

To factor do the following.

Step one:
Multiply 15 times -8
You get -120

Step two:
Find two numbers that multiply to -120 and add to -14
(-20)(6) = -120
-20+6 = -14

Step three:
Rewrite the original polynomial using the numbers you found in step two.
15a^2-20a+6a-8

Step four:
Factor by grouping
Find the GCF of the first two terms
15a^2-20a = 5a(3a-4)
Find the GCF of the last two terms
6a-8 = 2(3a-4)

Step five:
Factor out the common stuff in the parenthesis.
5a(3a-4)+2(3a-4) = (3a-4)(5a+2)

And now you have the completely factored form of the original polynomial.

Hope this helps. :)

Answer 5

split -14 into 2 parts that product is product of 15 and -8 that is -120

by trial and error we get -20* 6
now 15a^2-14a-8
= 15a^2 - 20a + 6a -8
= 5a(3a-4)+2(3a-4)
= (3a-4)(5a+2)

Answer 6

Sometimes it is a matter of just trial and error.

Start with the 8. Which numbers multiplied together equal -8??
I thought of
1, -8
2, -4
4, -2
8, -1

Now look at the 15 and do the same
1, 15
3, 5

Now try different combinations until you get the right one.
Pay attention to your signs
(5a +2)(3a -4)

Answer 7

solve the quadratic equation 15a^2 - 14a - 8 = 0.
find a1, a2.

this will then factorize to (a - a1)(a-a2).

Answer 8

15*-8=-120
first you need to choose factors
of -120 such that there sum is-14
-20*6
-20a+6a=-14a
Now it is easy
15a^2-14a-8
=15a^2-20a+6a-8
=5a[3a-4]+2[3a-4]
=[5a+2][3a-4]

Answer 9

15a^2 - 14a -8
15a^2 - 20a + 6a - 8
5a(3a - 4) + 2 (3a - 4)
=(3a - 4)(5a + 2)

Answer 10

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