Solving a quadratic with a variable and constant

Question

For example. Solve the following equation.

$$3x^2 - 22 a x + 24a^2 = 0,$$

where $a$ is positive. Does this make use of the discriminant? If so, how? Thanks for the help!

Answer 1

$$3x^2-22ax+24a^2=3x^2-18ax-4ax+24a^2=$$ $$=3x(x-6a)-4a(x-6a)=(x-6a)(3x-4a),$$ which gives the answer $$\left\{6a,\frac{4}{3}a\right\}$$

Answer 2

Well, if we want to solve for $x$ we can use:

$$\text{a}\cdot x^2+\text{b}\cdot x+\text{c}=0\space\Longleftrightarrow\space x=\frac{-\text{b}\pm\sqrt{\text{b}^2-4\cdot\text{a}\cdot\text{c}}}{2\cdot\text{a}}\tag1$$

So, when we have:

$$3\cdot x^2+\left(-22\text{a}\right)\cdot x+\left(24\text{a}^2\right)=0\space\Longleftrightarrow\space$$ $$x=\frac{-\left(-22\text{a}\right)\pm\sqrt{\left(-22\text{a}\right)^2-4\cdot3\cdot\left(24\text{a}^2\right)}}{2\cdot3}=\frac{22\text{a}\pm\sqrt{\color{red}{196\text{a}^2}}}{6}\tag2$$