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Maths-

General

Easy

Question

Solve the radical equation $square root of 6 x minus 20 end root minus x equals negative 6$ . Check for extraneous solutions

hint Hint:

Rearrange the equation and then solve for x.

The correct answer is: x = 4

Complete step by step solution:
Here we have the equation $square root of 6 x minus 20 end root minus x equals negative 6$
On rearranging, we have $square root of 6 x minus 20 end root equals x minus 6$
On squaring, we have $6 x minus 20 equals left parenthesis x minus 6 right parenthesis squared$
$not stretchy rightwards double arrow 6 x minus 20 equals x squared minus 12 x plus 36$
$not stretchy rightwards double arrow x squared minus 12 x minus 6 x plus 36 plus 20 equals 0$
$not stretchy rightwards double arrow x squared minus 18 x plus 56 equals 0$
Now, this is a quadratic equation with a = 1, b = -18, c = 56
Roots can be found with, $x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction$
$x equals fraction numerator 18 plus-or-minus square root of 324 minus 4 cross times 1 cross times 56 end root over denominator 2 cross times 1 end fraction$
On solving, we get x = 14, 4
On substituting the value of x in the equation, we get
$square root of 6 cross times 14 minus 20 end root minus 14 equals square root of 64 minus 14 equals 8 minus 14 equals negative 6$
Hence x = 14 works.
Now, we have $square root of 6 cross times 4 minus 20 end root minus 4 equals square root of 4 minus 4 equals 2 minus 4 equals negative 2 not equal to negative 6$
Hence x = 4 does not work.
Hence x = 4 is an extraneous solution.

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