Question

Simplify completely (7x²+11x-6)/(7x²-10x+3)

Answer 1

One way of simplifying the given expression is by factoring them, and then simplifying the common terms.

So, we need to write:

$\begin{gathered} 7x^{2}+11x-6=7(x-a)(x-b) \\ \text{and} \\ 7x^{2}-10x+3=7(x-c)(x-d) \end{gathered}$

The constants a and b are the zeros of the first expression, and the constants c and d are the zeros of the second expression.

So, we can find those zeros using the quadratic formula. We obtain, for the first expression:

$\begin{gathered} x=\frac{-11\pm\sqrt[]{11^{2}-4(7)(-6)}}{2(7)} \\ \\ x=\frac{-11\pm\sqrt[]{289}}{14} \\ \\ x=\frac{-11\pm17}{14} \\ \\ a=\frac{-11-17}{14}=-2 \\ \\ b=\frac{-11+17}{14}=\frac{6}{14}=\frac{3}{7} \end{gathered}$

And, for the second expression, we obtain:

$\begin{gathered} x=\frac{-(-10)\pm\sqrt[]{(-10)^{2}-4(7)(3)}}{2(7)} \\ \\ x=\frac{10\pm\sqrt[]{16}}{14} \\ \\ x=\frac{10\pm4}{14} \\ \\ c=\frac{10-4}{14}=\frac{6}{14}=\frac{3}{7} \\ \\ d=\frac{10+4}{14}=1 \end{gathered}$

Then, we can write:

$\begin{gathered} 7x^2+11x-6=7(x-(-2))(x-\frac{3}{7})=7(x+2)(x-\frac{3}{7}) \\ \\ 7x^2-10x+3=7(x-\frac{3}{7})(x-1) \end{gathered}$

Thus, the given function can be simplified as follows:

$\frac{7x²+11x-6}{7x²-10x+3}=\frac{7(x+2)(x-\frac{3}{7})}{7(x-\frac{3}{7})(x-1)}=\frac{x+2}{x-1}$

Therefore, the answer is:

$\mathbf{\frac{x+2}{x-1}}$