Question

solve
\\(\sqrt{y + 15} - \sqrt{y - 15} = 1\\)

select the correct choice below and fill in any answer boxes present in your choice.

\\(\bigcirc\\) a. \\(y = \square\\)
(use a comma to separate answers as needed. type an integer or a simplified fraction.)
\\(\bigcirc\\) b. there is no solution.

Explanation:

Step1: Isolate one square root

Given the equation $\sqrt{y + 15}-\sqrt{y - 15}=1$, we first isolate one of the square roots. Let's move $\sqrt{y - 15}$ to the right side:
$\sqrt{y + 15}=1+\sqrt{y - 15}$

Step2: Square both sides

Now, square both sides of the equation to eliminate the square roots. Remember that $(a + b)^2=a^2 + 2ab + b^2$. So,
$(\sqrt{y + 15})^2=(1+\sqrt{y - 15})^2$
$y + 15=1+2\sqrt{y - 15}+(y - 15)$

Step3: Simplify the equation

Simplify the right - hand side:
$y + 15=1+y - 15+2\sqrt{y - 15}$
$y + 15=y - 14+2\sqrt{y - 15}$
Subtract $y$ from both sides:
$15=- 14+2\sqrt{y - 15}$

Step4: Isolate the square root term

Add 14 to both sides:
$15 + 14=2\sqrt{y - 15}$
$29=2\sqrt{y - 15}$

Step5: Solve for the square root

Divide both sides by 2:
$\sqrt{y - 15}=\frac{29}{2}$

Step6: Square both sides again

Square both sides to solve for $y$:
$y-15 = (\frac{29}{2})^2=\frac{841}{4}$

Step7: Solve for y

Add 15 to both sides. Since $15=\frac{60}{4}$, we have:
$y=\frac{841}{4}+\frac{60}{4}=\frac{901}{4}$

Step8: Check the solution

We need to check if $y = \frac{901}{4}$ satisfies the original equation.
First, $\sqrt{y + 15}=\sqrt{\frac{901}{4}+15}=\sqrt{\frac{901 + 60}{4}}=\sqrt{\frac{961}{4}}=\frac{31}{2}$
Second, $\sqrt{y - 15}=\sqrt{\frac{901}{4}-15}=\sqrt{\frac{901-60}{4}}=\sqrt{\frac{841}{4}}=\frac{29}{2}$
Then, $\sqrt{y + 15}-\sqrt{y - 15}=\frac{31}{2}-\frac{29}{2}=\frac{2}{2}=1$, which matches the original equation.

Answer:

A. $y=\frac{901}{4}$