Find the value of $x$ for which the equations $f (x) = 3 x^{2} - 1$ and $g (x) = 3 + x$ are equal.

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Solution

Compute the required value:
Given: $f (x) = 3 x^{2} - 1$ and $g (x) = 3 + x$
Equating $f (x)$ and $g (x)$
$\Rightarrow f (x) = g (x) \Rightarrow 3 x^{2} - 1 = 3 + x \Rightarrow 3 x^{2} - x - 1 - 3 = 0 \Rightarrow 3 x^{2} - x - 4 = 0 \Rightarrow 3 x^{2} + 3 x - 4 x - 4 = 0 \Rightarrow 3 x (x + 1) - 4 (x + 1) = 0 \Rightarrow (x + 1) (3 x - 4) = 0$
$\Rightarrow$ $x = - 1, \frac{4}{3}$
Hence, the value of $x$ at which $f (x)$ and $g (x)$ are equal are $\frac{4}{3}$ and $- 1$ .

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