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Maths-

General

Easy

Question

Solve the radical equation $x plus 2 equals square root of x plus 2 end root$ . Identify any extraneous solutions

hint Hint:

Rearrange the equation and then solve for x.

The correct answer is: x = -1

Complete step by step solution:
Here given equation is $x plus 2 equals square root of x plus 2 end root$
On squaring both the sides, we have $left parenthesis x plus 2 right parenthesis squared equals left parenthesis square root of x plus 2 end root right parenthesis squared$
$table attributes columnspacing 1em end attributes row cell space space space space space space space not stretchy rightwards double arrow x squared plus 4 x plus 4 equals x plus 2 end cell row cell not stretchy rightwards double arrow x squared plus 3 x plus 2 equals 0 end cell end table$
Now, this is a quadratic equation with a = 1, b = 3, c = 2
Roots can be found with, $x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction$
$x equals fraction numerator negative 3 plus-or-minus square root of 9 minus 4 cross times 1 cross times 2 end root over denominator 2 cross times 1 end fraction$
$x equals fraction numerator negative 3 plus-or-minus square root of 9 minus 8 end root over denominator 2 end fraction$
$not stretchy rightwards double arrow x equals fraction numerator negative 3 plus-or-minus square root of 1 over denominator 2 end fraction$
$not stretchy rightwards double arrow x equals fraction numerator negative 3 plus-or-minus 1 over denominator 2 end fraction$
$not stretchy rightwards double arrow x equals negative 2 text or end text minus 1$
Here, on substituting x = -2 or -1 in given equation we get
$negative 2 plus 2 equals square root of negative 2 plus 2 end root$
$not stretchy rightwards double arrow 0 equals square root of 0 equals 0$
Hence x = -2 works.
Now, $negative 1 plus 2 equals square root of negative 1 plus 2 end root$
$not stretchy rightwards double arrow 1 equals square root of 1 equals 1$
Here x = -1 also works.
Hence no extraneous solutions.

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