Solve the following quadratic equation for $x$: $\sqrt{3}\,x^2 + 10x + 7\sqrt{3} = 0$
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
$\sqrt{3}\,x^2 + 10x + 7\sqrt{3} = 0$
Splitting the middle term: $10x = 3x + 7x$
$$\sqrt{3}\,x^2 + 3x + 7x + 7\sqrt{3} = 0$$
$$\sqrt{3}\,x(\,x + \sqrt{3}\,) + 7(\,x + \sqrt{3}\,) = 0$$
$$(\,x + \sqrt{3}\,)(\sqrt{3}\,x + 7) = 0$$
$$\therefore\quad x = -\sqrt{3} \quad \text{or} \quad x = -\dfrac{7}{\sqrt{3}} = -\dfrac{7\sqrt{3}}{3}$$
Source: Chapter 4, Section 4.3 (Factorisation method)
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Explanation
- Split the middle term so that the product of the two parts equals $\sqrt{3} \times 7\sqrt{3} = 21$ and their sum is 10. Here $3 + 7 = 10$ and $3 \times 7 = 21$ ✓
- After grouping, equate each linear factor to zero to get the two roots.
- Rationalize $-7/\sqrt{3}$ by multiplying numerator and denominator by $\sqrt{3}$ to write it neatly as $-7\sqrt{3}/3$. Examiners award 1 mark for correct factorisation and 1 mark for both roots stated correctly.