# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5/7

Exercises: Try Out - Set 5: Real exam simulation to sharpen your skills and build confidence. - Problem 7

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## Exercise 7

### Question

export const metadata = {
  title: "Problem 7",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

If <InlineMath math="f(x) = 2x + 3" /> and <InlineMath math="3f^2(x) = f(x) + 7" /> are satisfied by <InlineMath math="x_1" /> and <InlineMath math="x_2" />, then the value of <InlineMath math="x_1 + x_2 =" /> ...


### Choices

- [ ] $$\frac{8}{3}$$
- [ ] $$-\frac{8}{3}$$
- [ ] $$\frac{17}{6}$$
- [x] $$-\frac{17}{6}$$
- [ ] $$\frac{19}{9}$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 7",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

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

#### Substituting the Function

The first step is to substitute <InlineMath math="f(x)" /> into the given equation.

<MathContainer>
  <BlockMath math="3(2x + 3)^2 = (2x + 3) + 7" />
</MathContainer>

#### Simplifying the Equation

Next, we expand the square and simplify the equation into the general quadratic form <InlineMath math="ax^2 + bx + c = 0" />.

<MathContainer>
  <BlockMath math="3(4x^2 + 12x + 9) = 2x + 10" />
  <BlockMath math="12x^2 + 36x + 27 = 2x + 10" />
</MathContainer>

Move all terms to the left side:

<MathContainer>
  <BlockMath math="12x^2 + 36x - 2x + 27 - 10 = 0" />
  <BlockMath math="12x^2 + 34x + 17 = 0" />
</MathContainer>

#### Using the Sum of Roots Formula

From the quadratic equation <InlineMath math="12x^2 + 34x + 17 = 0" />, we obtain the coefficients:
- <InlineMath math="a = 12" />
- <InlineMath math="b = 34" />
- <InlineMath math="c = 17" />

The sum of the roots of a quadratic equation can be calculated using the formula <InlineMath math="x_1 + x_2 = -\frac{b}{a}" />.

<MathContainer>
  <BlockMath math="x_1 + x_2 = -\frac{34}{12}" />
</MathContainer>

Simplify the fraction by dividing the numerator and denominator by <InlineMath math="2" />:

<MathContainer>
  <BlockMath math="x_1 + x_2 = -\frac{17}{6}" />
</MathContainer>

#### Conclusion

Thus, the value of <InlineMath math="x_1 + x_2" /> is <InlineMath math="-\frac{17}{6}" />.


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