Solve by factoring.

Question

6t+9=4t²⁺¹

Answer 1

When we're asked to solve by factoring, we think of quadratic equations. We have all the necessary components here to make one, they're just scattered. Let's first bring all the terms to one side, preferably to the right so our t² term remains positive. 

0=4t²-6t-9+1 -> 0=4t²-6t-8.

We now need to find all the factors of 8 and 4.

8: 1 and 8, 2 and 4.

4: 1 and 4, 2 and 2. 

After this, we can use OI of FOIL to figure out whether or not any combination of these will give us 6 when we add or subtract the O and I terms. i.e (2t  2)(2t  4) = 8t  4t. Any sign we put in between these two will not give us 6t. You can try the rest on your own but clearly this method won't work. So we move on to our back plan, the quadratic formula. A good way to memorize it is to sing it to the "Pop goes the weasel" tune.  But more to the point, this is the quadratic formula:

x=(-b±√b²-4ac)/2a

As you may know every quadratic formula of degree 2 will have the form 0=ax²+bx+c. This is where we get our a, b and c values. Now to plug them in and solve for x!

x=(6±√(-6)²-4(4)(-8))/2(4)

x=(6±√36+128)/8

x=(6±√164)/8. We can simplify this slightly by manipulating the √164. First we split it into √4√41 and then solve √4=2. Our result:

x=(6±2√41)/8. We can simplify further by dividing the top and bottom by 2.

x=(3±√41)/4 and that is our final answer! 

I hope this helped!

Answer 2

6t + 9 = 4t² + 1 ----> this should be your equation and not the one you posted because the one you posted would have to be solved through logarithmic methods which I am assuming you are not up to par with (especially based on the question you have asked).

Step 1: Whenever you hear "factoring" and you see you have same variables (like "t" in this case) having exponents of 2 or 1 (like 4t² and 6t) and you happen to also see whole numbers (like 9 and 1) jump to get all these on one side of the "=" expression.

Step 2: Which side do I pick, the right or the left? Which ever side that gets you a positive coefficient for the 4t². Note that if you move it to the left, it will be negative and if you keep it on the side that it is already in, you have a positive. So move everything to the right of the expression.

Step 3:

6t + 9 = 4t² + 1

-6t - 9 = - 9 - 6t

-------------------------------------

0 = 4t² - 8 - 6t

Step 4. Factoring

Now this is probably where you might have stumped because there are a variety of ways but there exists only one that will not waste your time and will not seem impossible. 

1. Get rid of the coefficient of t². 

    0 = 4t² - 6t -8

   ----------------

   0 = t² - (6/4)t - 2 

   which is simplified to:  0 = t² - (3/2)t -2

2. Now you want to make the above expression into factors. You note that it is hard, so you do complete the square. 

       2+ (9/16) = t² - (3/2)t + (9/16)

      The above expression simplifies to : (41/16) = (t - 3/4)²

3. plus or minus sqrt(41/16) = (t - 3/4)

4. Thus  t = +sqrt(41/16) + (3/4)          OR          t = -sqrt(41/16) + (3/4)

There you go, these are the two solutions. Now imagine had you done something else beside this and the quadratic formula. That would have been a pain to begin with.