Problem 7

Explanation

We are given the function $f(x) = 2x + 3$ and the equation $3f^2(x) = f(x) + 7$. We want to find the sum of the roots of the equation, which is $x_1 + x_2$.

Substituting the Function

The first step is to substitute $f(x)$ into the given equation.

Simplifying the Equation

Next, we expand the square and simplify the equation into the general quadratic form $ax^2 + bx + c = 0$.

Move all terms to the left side:

Using the Sum of Roots Formula

From the quadratic equation $12x^2 + 34x + 17 = 0$, we obtain the coefficients:

• $a = 12$
• $b = 34$
• $c = 17$

The sum of the roots of a quadratic equation can be calculated using the formula $x_1 + x_2 = -\frac{b}{a}$.

Simplify the fraction by dividing the numerator and denominator by $2$:

Conclusion

Thus, the value of $x_1 + x_2$ is $-\frac{17}{6}$.