Mathematics — CBSE Class 10 board question

$\sqrt{3}\,x^2 + 10x + 7\sqrt{3} = 0$

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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• 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.