Solve Quadratic Equation 24x² - 44x = -12 Without Division

Let's solve the given equation:

$24x^2-44x=-12$

$$First, let's arrange the equation by moving terms and combining like terms:

$24x^2-44x=-12 \\ 24x^2-44x+12 =0 \\$Now, instead of dividing both sides of the equation by the common factor of all terms in the equation (which is 4), we'll choose to factor it out of the parentheses:

$24x^2-44x+12 =0 \\ 4(6x^2-11x+3)=0$

From here we'll remember that the product of expressions will yield 0 only if at least one of the multiplying expressions equals zero,

however, the first factor in the expression we got is 4, which is obviously different from zero, therefore:

$6x^2-11x+3=0$

Now we notice that in the resulting equation the coefficient of the quadratic term is not 1, therefore, we'll solve the equation using the quadratic formula (let's recall it):

The rule states that for a quadratic equation in the general form:

$ax^2+bx+c =0$

there are two solutions (or fewer) which we can find using the formula:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$,

Let's continue and use the quadratic formula, noting that:

$\begin{cases} a=6\\ b=-11\\ c=3 \end{cases}$

therefore the solutions to the quadratic equation are:

$x_{1,2}=\frac{11\pm\sqrt{(-11)^2-4\cdot6\cdot3}}{2\cdot6} \\ x_{1,2}=\frac{11\pm\sqrt{121-72}}{12} \\ x_{1,2}=\frac{11\pm\sqrt{49}}{12}\\ x_{1,2}=\frac{11\pm7}{12}\\ x_1=\frac{11+7}{12}=\frac{18}{12}\\ x_2=\frac{11-7}{12}=\frac{4}{12}\\$Let's continue and reduce the fractions in the solutions we got:

$x_1=\frac{18}{12}=\frac{3\cdot\not{6}}{2\cdot\not{6}}\\ \boxed{x_1=\frac{3}{2}}\\$and

$x_2=\frac{4}{12}=\frac{\not{4}}{3\cdot\not{4}}\\ \boxed{x_2=\frac{1}{3}}\\$

Let's summarize the solution of the equation:

$24x^2-44x=-12 \\ 24x^2-44x+12 =0 \\ \\ \downarrow\\ 4(6x^2-11x+3)=0\\ \downarrow\\ 6x^2-11x+3=0 \\ \downarrow\\ x_{1,2}=\frac{11\pm\sqrt{49}}{12}\\ \downarrow\\ x_1=\frac{18}{12}\rightarrow\boxed{x_1=\frac{3}{2}}\\ x_2=\frac{4}{12}\rightarrow\boxed{x_2=\frac{1}{3}}\\ \downarrow\\ \boxed{x=\frac{3}{2},\frac{1}{3}}$

Therefore, the correct answer is answer D.