First slide

Theory of equations

Question

$The number of solutions of \sqrt{3 x^{2} + x + 5} = x - 3 is$

Moderate

A

0
B

1
C

2
D

4

Solution

$We have \sqrt{3 x^{2} + x + 5} = x - 3$
$We must have 3 x^{2} + x + 5 \geq 0 and x - 3 \geq 0 or x \geq 3$
$\sqrt{3 x^{2} + x + 5} = x - 3 - - - - - - - (i)$
Squaring both sides of (i), we get
$\begin{array}{l} 3 x^{2} + x + 5 = x^{2} - 6 x + 9 \\ \Rightarrow 2 x^{2} + 7 x - 4 = 0 \\ \Rightarrow (2 x - 1) (x + 4) = 0 \\ \Rightarrow x = 1 / 2, - 4 \end{array}$
$These values does not satisfy the inequality x \geq 3 .$
Thus, (i) has no solution.

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