Question

The line $y = 3$ intersects the graph of $y = 4x^2 + x - 1$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m - n$.

Answer 1

Answer:

61

Step-by-step explanation:

Let's find the points $A$ and $B$.

We know that the $y$-coordinates of both are $3$.

So let's first solve:

$3=4x^2+x-1$

Subtract 3 on both sides:

$0=4x^2+x-1-3$

Simplify:

$0=4x^2+x-4$

I'm going to use the quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, to solve.

We must first compare to the quadratic equation, $ax^2+bx+c=0$.

$a=4$

$b=1$

$c=-4$

$\frac{-1 \pm \sqrt{1^2-4(4)(-4)}}{2(4)}$

$\frac{-1 \pm \sqrt{1+64}}{8}$

$\frac{-1 \pm \sqrt{65}}{8}$

Since the distance between the points $A$ and $B$ is horizontal. We know this because they share the same $y-coordinate$.This means we just need to find the positive difference between the $x$-values we found for the points of $A$ and $B$.

So that is, the distance between $A$ and $B$ is:

$\frac{-1+\sqrt{65}}{8}-\frac{-1-\sqrt{65}}{8}$

$\frac{-1+\sqrt{65}+1+\sqrt{65}}{8}$

$\frac{2\sqrt{65}}{8}$

$\frac{\sqrt{65}}{4}$

If we compare this to $\frac{\sqrt{m}}{n}$, we should see that:

$m=65 \text{ and } n=4$.

So $m-n=65-4=61$.

Answer 2

Answer:

m - n = 61

Step-by-step explanation:

$y=4x^2+x-1\\\\y=3\\\\\text{Find the points A and B}.\\\\4x^2+x-1=3\qquad\text{subtract 3 from both sides}\\\\4x^2+x-1-3=3-3\\\\4x^2+x-4=0$

$\text{Use the quadratic formula:}\\\\a=4,\ b=1,\ c=-4\\\\b^2-4ac=1^2-4(4)(-4)=1+64=65\\\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\to x=\dfrac{-1\pm\sqrt{65}}{2(4)}=\dfrac{-1\pm\sqrt{65}}{8}$

$\text{Therefore}\\\\A\left(\dfrac{-1-\sqrt{65}}{8},\ 3\right),\ B\left(\dfrac{-1+\sqrt{65}}{8},\ 3\right)\\\\\text{The formula of a distance between two points:}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\\text{Substitute:}\\\\d=\sqrt{\left(\dfrac{-1+\sqrt{65}}{8}-\dfrac{-1-\sqrt{65}}{8}\right)^2+(3-3)^2}\\\\d=\sqrt{\left(\dfrac{-1+\sqrt{65}-(-1)-(-\sqrt{65})}{8}\right)^2+0^2}\\\\d=\sqrt{\left(\dfrac{-1+\sqrt{65}+1+\sqrt{65}}{8}\right)^2}\\\\d=\sqrt{\left(\dfrac{2\sqrt{65}}{8}\right)^2}$

$d=\sqrt{\left(\dfrac{\sqrt{65}}{4}\right)^2}\Rightarrow d=\dfrac{\sqrt{65}}{4}\Rightarrow\dfrac{\sqrt{65}}{4}=\dfrac{\sqrt{m}}{n}\\\\\text{Therefore}\ m=65\ \text{and}\ n=4\\\\m-n=65-4=61$