Question

solve for y, where y is a real number.\\(\sqrt{5y - 21} - \sqrt{y - 5} = 2\\)\\((\text{if there is more than one solution, separate them with commas.})\\)\\(y = \boxed{}\\)

Explanation:

Step1: Isolate one square root

Start with the equation $\sqrt{5y - 21} - \sqrt{y - 5} = 2$. Add $\sqrt{y - 5}$ to both sides to isolate $\sqrt{5y - 21}$:
$\sqrt{5y - 21} = 2 + \sqrt{y - 5}$

Step2: Square both sides

Square both sides to eliminate the square root on the left. Remember that $(a + b)^2 = a^2 + 2ab + b^2$:
$(\sqrt{5y - 21})^2 = (2 + \sqrt{y - 5})^2$
Simplify both sides:
$5y - 21 = 4 + 4\sqrt{y - 5} + (y - 5)$

Step3: Simplify the equation

Simplify the right side:
$5y - 21 = y - 1 + 4\sqrt{y - 5}$
Subtract $y$ and add $1$ to both sides:
$4y - 20 = 4\sqrt{y - 5}$
Divide both sides by $4$:
$y - 5 = \sqrt{y - 5}$

Step4: Let $u = \sqrt{y - 5}$

Let $u = \sqrt{y - 5}$, then $u^2 = y - 5$. Substitute into the equation:
$u^2 = u$
Subtract $u$ from both sides:
$u^2 - u = 0$
Factor:
$u(u - 1) = 0$
So $u = 0$ or $u = 1$

Step5: Solve for $y$

Case 1: $u = 0$
$\sqrt{y - 5} = 0$
Square both sides:
$y - 5 = 0$
$y = 5$

Case 2: $u = 1$
$\sqrt{y - 5} = 1$
Square both sides:
$y - 5 = 1$
$y = 6$

Step6: Check solutions

Check $y = 5$:
Left side: $\sqrt{5(5) - 21} - \sqrt{5 - 5} = \sqrt{25 - 21} - \sqrt{0} = \sqrt{4} - 0 = 2 - 0 = 2$
Right side: $2$
So $y = 5$ is a solution.

Check $y = 6$:
Left side: $\sqrt{5(6) - 21} - \sqrt{6 - 5} = \sqrt{30 - 21} - \sqrt{1} = \sqrt{9} - 1 = 3 - 1 = 2$
Right side: $2$
So $y = 6$ is a solution.

Answer:

5, 6