Please help me simplify this equation!?

Question

Please simplify this for me:

12x^2 - 8lx + l^2 = 0

The answers are meant to be: x = l/6 & x = l/2, I just don't know the steps to take to get here, please help!

Answer 1

12x^2 - 8lx + l^2 = 0
Using x = [-b±√(b²-4ac)]/2a
the general formula for solving quadratic eqns,
where a= the coefficient of x²
b= coefficient of x
c= constant

So for 12x^2 - 8lx + l^2 = 0,
a= 12, b=-8I, c=I²

x
={8I±√[(-8I)²-4(12)(I²)]}/2(12)
={8I±√[64I²-48I²]}/24
=[8I± √(16I²)]/24
=[8I±4I]/24
Hence,
x= (8I+4I)/24
= 12I/24
= I/2

or
x= (8I-4I)/24
= 4I/24
= I/6

Answer 2

Divide by 12, and get
x^2 - 2lx/3 + l^2/12 = 0

Use the rule (a - b)^2 = a^2 - 2ab + b^2 = 0 with a=x, b = l/3
x^2 - 2lx/3 + l^2/9 -ll^2/9 + l^2/12 = 0

(x - l/3)^2 - l^2/9 + l^2/12 = 0

(x-l/3)^2 - l^2/36 = 0

Now, use the law (a + b)(a - b) = a^2 - b^2
(x - l/3 - l/6)(x - l/3 + l/6) = 0
(x - l/2)(x - l/6) = 0

x1 = l/2
x2 = l/6

Answer 3

I keep confusing the I with a 1, so I will use y instead:

Let's try to factorize our equation first, assuming that it has two factors (ax+y) and (bx+y):
(ax+y)(bx+y) = 12x^2 - 8xy + y^2
ab x^2 + (a+b)xy + y^2 = 12x^2 - 8xy + y^2

So, we have:
ab = 12 and a+b = -8

You may right away see that a=-6 and b=-2 (or vice versa), or you may solve for a:

To solve for a, first observe that b=8-a
Then, substitute in the equation ab=12 to get
a(8-a)=12
8a - a^2 - 12 = 0
a^2 - 8a + 12 = 0

Using the quadratic formula would give a=-6 or a=-2


Now, we know the original equation factors into
(-6x+y)(-2x+y)=0

Since the product is equal to zero, at least one of the factors must be equal to zero. So,

-6x + y = 0
6x = y
x = y/6

or

-2x + y = 0
2x = y
x = y/2

Answer 4

That's just a simple quadratic equation.

Use:

x = [ -b ± sqrt( b^2 - 4ac ) ] / (2a)

Where a = 12, b = -8 and c = l^2. That equation will give you 2 solutions to x.

If you really need to use the "long way" then use complete-the-square method - which is really just the above equation. Make use of the identity:

x^2 +ax = (x + a/2)^2 - (a/2)^2

Answer 5

multiply it then by x

Answer 6

multiply it then by x