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Maths-

General

Easy

Question

Solve the radical equation $x equals square root of 7 x plus 8 end root$ . 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 $x equals square root of 7 x plus 8 end root$
On squaring both the sides, we have $x squared equals 7 x plus 8$
$not stretchy rightwards double arrow x squared minus 7 x minus 8 equals 0$
Now, this is a quadratic equation with a = 1, b = -7, c = -8
Roots can be found with, $x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction$
$x equals fraction numerator 7 plus-or-minus square root of 49 minus 4 cross times 1 cross times negative 8 end root over denominator 2 cross times 1 end fraction$
$x equals fraction numerator 7 plus-or-minus square root of 49 plus 32 end root over denominator 2 end fraction$
$not stretchy rightwards double arrow x equals fraction numerator 7 plus-or-minus square root of 81 over denominator 2 end fraction$
$not stretchy rightwards double arrow x equals fraction numerator 7 plus-or-minus 9 over denominator 2 end fraction$
$not stretchy rightwards double arrow x equals 8 text or end text minus 1$

Here, on substituting x = 8 or -1 in given equation we get
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 8 equals square root of 7 cross times 8 plus 8 end root end cell row cell not stretchy rightwards double arrow 8 equals square root of 64 equals 8 end cell end table$
Hence x = 8 works.
Now, $negative 1 equals square root of 7 cross times negative 1 plus 8 end root$
$not stretchy rightwards double arrow negative 1 not equal to square root of 1 equals 1$
Here the extraneous solution is x = -1

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