Question

1. graph the following quadratic equation and find the solution by interpreting the graph. state how many solutions there are (1, 2, or none). x² - 1 = 4

1
2
none

1. if the geometric mean of a and 30 is 9√10, find the value of a.

35
27
39
28
none of these answers are correct.

Explanation:

Step1: Rearrange the quadratic equation

First, rewrite $x^{2}-1 = 4$ as $x^{2}-5=0$. The graph of $y = x^{2}-5$ is a parabola of the form $y = ax^{2}+bx + c$ where $a = 1$, $b = 0$, $c=-5$. The vertex of the parabola $y=ax^{2}+bx + c$ is at $x=-\frac{b}{2a}=0$, and $y(0)=-5$. The parabola opens upwards since $a = 1>0$. To find the solutions, we set $y = 0$, so $x^{2}-5=0$, which gives $x=\pm\sqrt{5}$. The graph of $y=x^{2}-5$ intersects the $x -$axis at two points. So the number of solutions of the equation $x^{2}-1 = 4$ is 2.

Step2: Use geometric - mean formula

The geometric mean of two numbers $a$ and $b$ is $\sqrt{ab}$. Given that the geometric mean of $a$ and 30 is $9\sqrt{10}$, we have $\sqrt{30a}=9\sqrt{10}$. Square both sides of the equation: $30a=(9\sqrt{10})^{2}$. Since $(9\sqrt{10})^{2}=9^{2}\times(\sqrt{10})^{2}=81\times10 = 810$. Then $a=\frac{810}{30}=27$.

Answer:

1. B. 2
2. B. 27