Question

Show all work to solve the equation for x. If a solution is extraneous, be sure to identify it in your final answer.

square root of the quantity x minus 3 end quantity plus 5 equals x

Answer 1

x=7 is the solution for the given equation

x=4 is the extraneous solution .

Given :

$\sqrt{x-3} +5=x$

Explanation :

To solve the given equation for x , we need to remove the square root first

Subtract 5 on both sides

$\sqrt{x-3}+5=x\\ \sqrt{x-3}=x-5$

Take square on both sides to remove square root

$x-3=(x-5)^2\\x-3=x^{2} -10x+25$

Subtract x  and add 3 on both sides

$0=x^{2} -11x+28\\0=(x-7)(x-4)$

Set the factors =0 and solve for x

$x-7 =0\\x=7\\\\\\x-4=0\\x=4$

x values are 4  and 7

Now we check the values using original equation and verify it

$x=4\\\sqrt{x-3} +5=x\\\sqrt{4-3} +5=4\\\\6=4$

x=4 does not satisfies the original equation

Now we check x=7

$\sqrt{7-3} +5=7\\7=7$

x=7 is the solution  

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