Question

Solve the following equation and check for extraneous solutions. Show all work. Thank you.

Answer 1

`x=19`

Explanation:

1) Use this rule:
`(x^a)^b=x^a^b` .

`(x-3)^{(1*1)/(2*2) } =√(x-15)`

2) Simplify 1 * 1 to 1.

`\sqrt[2*2]{x-3} =√(x-15)`

3) Simplify 2 * 2 to 4.

`\sqrt[4]{x-3} =√(x-15)`

4) Square both sides.

`√(x-3) =x-15`

5) Square both sides.

`x-3=x^2-30x+225`

6) Move all terms to one side.

`x-3-x^2+30x-225=0`

7) Simplify
`x-3-x^2+30x-225` to
`31x-228-x^2`.

`31x-228-x^2=0`

8) Multiply both sides by -1.

`x^2-31x+228=0`

9) Factor
`x^2-31x+228`.

1) Ask: Which two numbers add up to -31 and multiply to 228?

`-19` and
`-12`

2) Rewrite the expression using the above.

`(x-19)(x-12)`

10) Solve for
`x`.

1) Ask when will
`(x-19)(x-12)` equal zero?

When
`x-19 =0` or
`x-12=0`

2) Solve each of the 2 equations above.

`x=19,12`

11) Check solution.

When
`x=12` 2, the original equation
`\sqrt{√(x-3)}=√(x-15)` does not hold true. We will drop
`x=12` from the solution set.

12) Therefore,

`x=19`

Check the Answer:

`\sqrt{√(x-3)}=√(x-15)`

1) Let
`x=19`.

`\sqrt{√(19-3)}=√(19-15)`

2) Simplify 19 - 3 to 16.

`\sqrt{√(16)}=√(19-15)`

3) Since 4 * 4 is 16 6, the square root of 16 is 4.

`√(4)=√(19-15)`

4) Since 2 * 2 = 4, the square root of 4 is 2.

`2=√(19-15)`

5) Simplify 19 - 15 to 4.

`2=√(4)`

6) Since 2 * 2 = 4, the square root of 4 is 2.

`2 = 2`

Answer 2

Answer: 19

Hope this helps.