Question

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

Answer 1

Answer: 19

Hope this helps.

Answer 2

Answer:

$x=19$

Step-by-step explanation:

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

$(x-3)^{\frac{1*1}{2*2} } =\sqrt{x-15}$

2) Simplify 1 * 1 to 1.

$\sqrt[2*2]{x-3} =\sqrt{x-15}$

3) Simplify 2 * 2 to 4.

$\sqrt[4]{x-3} =\sqrt{x-15}$

4) Square both sides.

$\sqrt{x-3} =x-15$

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 $\sqrt{\sqrt{x-3}}=\sqrt{x-15}$ does not hold true. We will drop $x=12$ from the solution set.

12) Therefore,

$x=19$

Check the Answer:

$\sqrt{\sqrt{x-3}}=\sqrt{x-15}$

1) Let $x=19$.

$\sqrt{\sqrt{19-3}}=\sqrt{19-15}$

2) Simplify 19 - 3 to 16.

$\sqrt{\sqrt{16}}=\sqrt{19-15}$

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

$\sqrt{4}=\sqrt{19-15}$

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

$2=\sqrt{19-15}$

5)  Simplify  19 - 15  to  4.

$2=\sqrt{4}$

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

$2 = 2$