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Maths-

General

Easy

Question

Solve the radical equation $square root of 4 x plus 5 end root minus square root of x plus 1 end root equals 1$ Check for extraneous solutions

hint Hint:

Rearrange the equation and then solve for x.

The correct answer is: Hence no extraneous solution

Complete step by step solution:
Here we have the equation $square root of 4 x plus 5 end root minus square root of x plus 1 end root equals 1$
$not stretchy rightwards double arrow square root of 4 x plus 5 end root equals 1 plus square root of x plus 1 end root$
On squaring both the sides, we have $4 x plus 5 equals left parenthesis 1 plus square root of x plus 1 end root right parenthesis left parenthesis 1 plus square root of x plus 1 end root right parenthesis$
$not stretchy rightwards double arrow 4 x plus 5 equals 1 plus 2 square root of x plus 1 end root plus x plus 1$
On subtracting x and 2 on both the sides, we get
$not stretchy rightwards double arrow 3 x plus 3 equals 2 square root of x plus 1 end root$
On squaring both the sides, we get
$9 x squared plus 18 x plus 9 equals 4 x plus 4$
$not stretchy rightwards double arrow 9 x squared plus 14 x plus 5 equals 0$
Now, this is a quadratic equation with a = 9, b = 14, c = 5
Roots can be found with, $x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction$
$x equals fraction numerator negative 14 plus-or-minus square root of 196 minus 4 cross times 9 cross times 5 end root over denominator 2 cross times 9 end fraction$
On solving, we get $x equals negative 1 comma fraction numerator negative 5 over denominator 9 end fraction$
Here, on substituting $x equals negative 1 text or end text fraction numerator negative 5 over denominator 9 end fraction$ in given equation we get
$square root of 4 cross times negative 1 plus 5 end root minus square root of negative 1 plus 1 end root equals 1$
$not stretchy rightwards double arrow square root of 1 minus square root of 0 equals 1$
Hence x = - 1 works.
Now, $square root of 4 cross times negative 5 over 9 plus 5 end root minus square root of fraction numerator negative 5 over denominator 9 end fraction plus 1 end root equals 1$
$not stretchy rightwards double arrow square root of 25 over 9 end root minus square root of 4 over 9 end root equals 5 over 3 minus 2 over 3 equals 3 over 3 equals 1$
Here $x equals negative 5 over 9$ also works.
Hence no extraneous solution

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