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zhannawk [14.2K]
2 years ago
12

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 w

ritten 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 $m - n$.
Mathematics
2 answers:
raketka [301]2 years ago
5 0

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.

Sergeu [11.5K]2 years ago
3 0

Answer:

<h2>m - n = 61</h2>

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

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