Simplify the rational expression.
\[
\frac{3 x^{2}-16 x-12}{x^{2}-9 x+18}
\]
Answer
Check the result by substituting a value for x into the original and simplified expressions and seeing if they give the same result. For example, if we let x = 1, the original expression gives \(\frac{3*1^{2}-16*1-12}{1^{2}-9*1+18} = -25/10 = -2.5\) and the simplified expression gives \(\frac{3*1+2}{1-3} = -5/2 = -2.5\). Since they are the same, the simplification is correct.
Step 1 ：Factorize the numerator \(3x^{2}-16x-12\) by grouping. Look for two numbers that multiply to \(3*(-12) = -36\) and add to -16. These numbers are -18 and 2. So rewrite -16x as -18x + 2x and group: \(3x^{2}-16x-12 = 3x^{2}-18x+2x-12 = 3x(x-6)+2(x-6) = (3x+2)(x-6)\)
Step 2 ：Factorize the denominator \(x^{2}-9x+18\) by looking for two numbers that multiply to 18 and add to -9. These numbers are -6 and -3. So we have: \(x^{2}-9x+18 = (x-6)(x-3)\)
Step 3 ：Substitute the factorized numerator and denominator into the original expression: \(\frac{3 x^{2}-16 x-12}{x^{2}-9 x+18} = \frac{(3x+2)(x-6)}{(x-6)(x-3)}\)
Step 4 ：Cancel out the common factor of (x-6) from the numerator and the denominator: \(\frac{(3x+2)(x-6)}{(x-6)(x-3)} = \frac{3x+2}{x-3}\)
Step 5 ：\(\boxed{\frac{3x+2}{x-3}}\) is the simplified form of the rational expression.
Step 6 ：Check the result by substituting a value for x into the original and simplified expressions and seeing if they give the same result. For example, if we let x = 1, the original expression gives \(\frac{3*1^{2}-16*1-12}{1^{2}-9*1+18} = -25/10 = -2.5\) and the simplified expression gives \(\frac{3*1+2}{1-3} = -5/2 = -2.5\). Since they are the same, the simplification is correct.
