# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-6/14

Exercises: Try Out - Set 6: Real exam simulation to sharpen your skills and build confidence. - Problem 14

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## Exercise 14

### Question

export const metadata = {
  title: "Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

If <InlineMath math="g(x) = 3x + 2" /> and <InlineMath math="2g^2(x) - g(x^2) - 5g(x) = 7" /> are satisfied by <InlineMath math="x_1" /> and <InlineMath math="x_2" />, then the value of <InlineMath math="x_1 + x_2" /> is ...


### Choices

- [x] $$-\frac{3}{5}$$
- [ ] $$\frac{1}{5}$$
- [ ] $$\frac{4}{3}$$
- [ ] $$\frac{2}{3}$$
- [ ] $$\frac{3}{5}$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the function <InlineMath math="g(x) = 3x + 2" />. We will substitute this function into the given equation.

<MathContainer>
  <BlockMath math="2g^2(x) - g(x^2) - 5g(x) - 7 = 0" />
</MathContainer>

First, determine the forms of <InlineMath math="g^2(x)" /> and <InlineMath math="g(x^2)" />:

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

Substitute these forms into the initial equation:

<MathContainer>
  <BlockMath math="2(9x^2 + 12x + 4) - (3x^2 + 2) - 5(3x + 2) - 7 = 0" />
  <BlockMath math="(18x^2 + 24x + 8) - (3x^2 + 2) - (15x + 10) - 7 = 0" />
</MathContainer>

Group like terms:

<MathContainer>
  <BlockMath math="(18 - 3)x^2 + (24 - 15)x + (8 - 2 - 10 - 7) = 0" />
  <BlockMath math="15x^2 + 9x - 11 = 0" />
</MathContainer>

This quadratic equation has roots <InlineMath math="x_1" /> and <InlineMath math="x_2" />.
The sum of the roots of a quadratic equation <InlineMath math="ax^2 + bx + c = 0" /> is <InlineMath math="x_1 + x_2 = -\frac{b}{a}" />.

<MathContainer>
  <BlockMath math="x_1 + x_2 = -\frac{9}{15}" />
  <BlockMath math="x_1 + x_2 = -\frac{3}{5}" />
</MathContainer>

Thus, the value of <InlineMath math="x_1 + x_2" /> is <InlineMath math="-\frac{3}{5}" />.


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