First slide

Theory of equations

Question

$Number of real solutions of \sqrt{2 x - 4} - \sqrt{x + 5} = 1 is$

Moderate

A

0
B

1
C

2
D

infinite

Solution

$We have \sqrt{2 x - 4} = 1 + \sqrt{x + 5} Squaring on both sides$
$\begin{array}{l} 2 x - 4 = 1 + (x + 5) + 2 \sqrt{x + 5} \\ \Rightarrow x - 10 = 2 \sqrt{x + 5} \\ s q u a r r i n g a g a i n \\ \Rightarrow x^{2} + 100 - 20 x = 4 x + 20 \\ \Rightarrow x^{2} - 24 x + 80 = 0 \end{array}$
$\begin{array}{l} \Rightarrow x = 4, 20 \\ Putting x = 4, we get \sqrt{4} - \sqrt{9} = 1, which is not possible \end{array}$
$Putting x = 20, we get \sqrt{36} - \sqrt{25} = 1$
$Hence, x = 20 is the only solution.$

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