Solve the quadratic equation 12x^2 + 36a^2= 43ax by factorization

Show your work

I prefer to write it as

12x^2 -43 ax + 36a^2 = 0

Look for something of the form

(4x -ma)(3x - na) where mn = 36 and
(3m + 4n) = 43

How about m = 9 and n = 4?
(4m -9a)(3m - 4a) It works

To solve the quadratic equation 12x^2 + 36a^2 = 43ax by factorization, we need to express the equation in the form of (x - p)(x - q) = 0, where p and q are the factors of the equation.

First, let's rearrange the equation to one side:
12x^2 + 36a^2 - 43ax = 0

Next, we'll look for common factors to simplify the equation:
12(x^2 + 3a^2) - 43ax = 0

Now, let's factor out the common factor of 12:
12(x^2 + 3a^2 - (43/12)ax) = 0

At this stage, we take note of the expression inside the parentheses: x^2 + 3a^2 - (43/12)ax.
To factorize this expression, we're looking for two numbers, p and q, such that their sum is -((43/12)a) and their product is (3a^2).

Given this, we can use the quadratic formula to find the factors:
x = (-b ± √(b^2 - 4ac)) / (2a)

Using the quadratic formula, we have:
x = (-(43/12)ax ± √((43/12)ax)^2 - 4(1)(3a^2)) / (2(1))

Simplifying further:
x = (-43ax/24 ± √((1849a^2x^2) / 144) - (12a^2)) / 2

Now, let's simplify the square root term:
√((1849a^2x^2) / 144) = √(1849a^2x^2) / √144
= (43ax) / 12

Our equation for x becomes:
x = (-43ax/24 ± (43ax) / 12 - (12a^2)) / 2

Simplifying:
x = -43ax/24 ± 43ax/24 - 12a^2/2

Combining like terms:
x = -43ax/24 ± 43ax/24 - 6a^2

Now, let's separate the two possible solutions:
x = (-43ax ± 43ax - 48a^2) / 24

Simplifying further:
x = (-48a^2) / 24
OR
x = (0) / 24

Finally, we have the two possible solutions for x:
x = -2a
OR
x = 0

Therefore, by factorization, the quadratic equation 12x^2 + 36a^2 = 43ax can be solved as x = -2a or x = 0.