Key Concepts and Formulas

• Solving Rational Equations: Identifying restrictions on the domain and manipulating the equation to a simpler form.
• Quadratic Equations: Determining the nature of roots using the discriminant ($\Delta = b^2 - 4ac$).
• Sum of Roots: For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$.

Step-by-Step Solution

Step 1: Analyze the Equation and Determine the Domain

We are given the equation: $\frac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \frac{5x^2 - 7x + 19}{3x^2 + 5x + 12}$ First, we need to ensure that the denominators are not equal to zero. Let's analyze each denominator.

• Denominator 1: $x^2 + 3x + 10$ To check for real roots, we compute the discriminant: $\Delta = b^2 - 4ac = (3)^2 - 4(1)(10) = 9 - 40 = -31$. Since $\Delta < 0$, $x^2 + 3x + 10$ has no real roots and is always positive.

• Denominator 2: $3x^2 + 5x + 12$ Similarly, we compute the discriminant: $\Delta = b^2 - 4ac = (5)^2 - 4(3)(12) = 25 - 144 = -119$. Since $\Delta < 0$, $3x^2 + 5x + 12$ has no real roots and is always positive.

Since both denominators are always positive and never zero for any real $x$, the domain is all real numbers.

Step 2: Simplify the Equation

Let's examine the difference between the numerators and denominators:

• $N_1 - D_1 = (3x^2 - 9x + 17) - (x^2 + 3x + 10) = 2x^2 - 12x + 7$
• $N_2 - D_2 = (5x^2 - 7x + 19) - (3x^2 + 5x + 12) = 2x^2 - 12x + 7$ We observe that $N_1 - D_1 = N_2 - D_2$. Let $P(x) = 2x^2 - 12x + 7$. So, we can rewrite the numerators as: $N_1 = D_1 + P(x)$ $N_2 = D_2 + P(x)$

Substituting these back into the original equation $\frac{N_1}{D_1} = \frac{N_2}{D_2}$: $\frac{D_1 + P(x)}{D_1} = \frac{D_2 + P(x)}{D_2}$ Separating the fractions: $1 + \frac{P(x)}{D_1} = 1 + \frac{P(x)}{D_2}$ Subtracting 1 from both sides: $\frac{P(x)}{D_1} = \frac{P(x)}{D_2}$ $\frac{2x^2 - 12x + 7}{x^2 + 3x + 10} = \frac{2x^2 - 12x + 7}{3x^2 + 5x + 12}$

Step 3: Solve the Simplified Equation

The equation is now in the form $\frac{A}{B} = \frac{A}{D}$, where $A = 2x^2 - 12x + 7$, $B = x^2 + 3x + 10$, and $D = 3x^2 + 5x + 12$. This equation holds true if either $A = 0$ or $B = D$.

• Case 1: The numerator is zero ($A = 0$) $2x^2 - 12x + 7 = 0$ The sum of the roots is given by $-\frac{b}{a} = -\frac{-12}{2} = 6$. The discriminant is $\Delta = (-12)^2 - 4(2)(7) = 144 - 56 = 88 > 0$, so the roots are real.

• Case 2: The denominators are equal ($B = D$) $x^2 + 3x + 10 = 3x^2 + 5x + 12$ $0 = 2x^2 + 2x + 2$ $0 = x^2 + x + 1$ The discriminant is $\Delta = (1)^2 - 4(1)(1) = 1 - 4 = -3 < 0$, so there are no real roots.

Step 4: Conclusion

The only real values of $x$ that satisfy the original equation are the roots of $2x^2 - 12x + 7 = 0$. The sum of these roots is 6.

Common Mistakes & Tips

• Always check the discriminant to ensure the roots are real before calculating the sum of the roots.
• Don't forget to consider both cases when solving equations of the form $\frac{A}{B} = \frac{A}{D}$: $A=0$ or $B=D$.
• Look for relationships between numerators and denominators to simplify the equation.

Summary

We simplified the given rational equation by recognizing a common difference between the numerators and denominators. This led to two cases: the numerator being zero or the denominators being equal. Only the case where the numerator was zero yielded real roots, and their sum was found to be 6.

Final Answer

The final answer is \boxed{6}.