B3. The expression is $\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}$.

a) Rationalize the denominator and simplify the expression.

b) Calculate the value of the expression for $x=\frac{5}{4}$.

## 21st Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, May 15, 2021

## Problems for 3rd Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Write your answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-10.jpg?height=174&width=810&top_left_y=584&top_left_x=635)
## B3.

a) If we multiply the numerator and the denominator of the expression $\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}$ by $\sqrt{x+1}+\sqrt{x-1}$ and simplify, we get $\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}} \cdot \frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}=\frac{(\sqrt{x+1}+\sqrt{x-1})^{2}}{(\sqrt{x+1})^{2}-(\sqrt{x-1})^{2}}=\frac{x+1+2 \sqrt{x+1} \sqrt{x-1}+x-1}{x+1-(x-1)}=\frac{2 x+2 \sqrt{(x+1)(x-1)}}{2}=$ $x+\sqrt{x^{2}-1}$.

b) If we substitute $x=\frac{5}{4}$ into the rationalized expression, we get $\frac{5}{4}+\sqrt{\left(\frac{5}{4}\right)^{2}-1}=\frac{5}{4}+\sqrt{\frac{25}{16}-\frac{16}{16}}=\frac{5}{4}+\sqrt{\frac{9}{16}}=\frac{5}{4}+\frac{3}{4}=$ $\frac{8}{4}=2$. Alternatively, we can substitute $x=\frac{5}{4}$ into the given expression: $\frac{\sqrt{\frac{5}{4}+1}+\sqrt{\frac{5}{4}-1}}{\sqrt{\frac{5}{4}+1}-\sqrt{\frac{5}{4}-1}}=\frac{\sqrt{\frac{9}{4}}+\sqrt{\frac{1}{4}}}{\sqrt{\frac{9}{4}}-\sqrt{\frac{1}{4}}}=\frac{\frac{3}{2}+\frac{1}{2}}{\frac{3}{2}-\frac{1}{2}}=\frac{\frac{4}{2}}{\frac{2}{2}}=2$.

Multiplying the given expression by $\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}$ ... 1 point

Simplifying the numerator to the form $x+1+2 \sqrt{x+1} \sqrt{x-1}+x-1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ ... 1 point

Simplifying the expression to the form $\frac{2 x+2 \sqrt{(x+1)(x-1)}}{2}=x+\sqrt{x^{2}-1} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1+1$ points

Calculating the value of the expression $x+\sqrt{x^{2}-1}=\frac{5}{4}+\sqrt{\left(\frac{5}{4}\right)^{2}-1}=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1+1$ points

## 21st Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, May 15, 2021  Solutions to Problems for 3rd Year

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-22.jpg?height=137&width=220&top_left_y=477&top_left_x=929)