Question

find the equation of all vertical asymptotes of the following function.
f(x)=\frac{x + 7}{sqrt{-x^{2}+4x + 5}}

Explanation:

Step1: Recall vertical - asymptote condition

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non - zero. Set the denominator equal to zero: $\sqrt{-x^{2}+4x + 5}=0$. Since the square root function $y = \sqrt{u}$ is zero when $u = 0$, we set $-x^{2}+4x + 5=0$.

Step2: Solve the quadratic equation

The quadratic equation $-x^{2}+4x + 5=0$ can be rewritten as $x^{2}-4x - 5=0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 1$, $b=-4$, and $c=-5$. Then $x=\frac{4\pm\sqrt{(-4)^{2}-4\times1\times(-5)}}{2\times1}=\frac{4\pm\sqrt{16 + 20}}{2}=\frac{4\pm\sqrt{36}}{2}=\frac{4\pm6}{2}$.

Step3: Calculate the roots

$x_1=\frac{4 + 6}{2}=5$ and $x_2=\frac{4-6}{2}=-1$. We also need to check that the numerator $x + 7$ is non - zero at these values. When $x=-1$, $x + 7=-1 + 7=6
eq0$. When $x = 5$, $x+7=5 + 7=12
eq0$.

Answer:

$x=-1,x = 5$