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

61

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=(-b\pm √(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`

`(-1 \pm √(1^2-4(4)(-4)))/(2(4))`

`(-1 \pm √(1+64))/(8)`

`(-1 \pm √(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:

`(-1+√(65))/(8)-(-1-√(65))/(8)`

`(-1+√(65)+1+√(65))/(8)`

`(2√(65))/(8)`

`(√(65))/(4)`

If we compare this to
`(√(m))/(n)`, we should see that:

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

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

Answer 2

m - n = 61

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=(-b\pm√(b^2-4ac))/(2a)\to x=(-1\pm√(65))/(2(4))=(-1\pm√(65))/(8)`

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

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