# Question #0dcca

Oct 13, 2016

$17 \sqrt{5}$

#### Explanation:

First, let's find the two points at which the two equations intersect.

From the first equation, we have $2 x - y = 7 \implies y = 2 x - 7$.

If we substitute this into the quadratic, we get

${\left(2 x - 7\right)}^{2} - x \left(x + 2 x - 7\right) = 11$

$\implies 4 {x}^{2} - 28 x + 49 - 3 {x}^{2} + 7 x = 11$

$\implies {x}^{2} - 21 x + 38 = 0$

$\implies \left(x - 2\right) \left(x - 19\right) = 0$

$\implies x = 2 \mathmr{and} x = 19$

We now have our two $x$-coordinates of the intersections as ${x}_{1} = 2$ and ${x}_{2} = 19$. We can substitute these into the first equation to get the points:

${y}_{1} = 2 \left(2\right) - 7 = - 3$

${y}_{2} = 2 \left(19\right) - 7 = 31$

So, we have the two points $A \left(2 , - 3\right)$ and $B \left(19 , 31\right)$. We can now find the distance between them using the formula

${\text{distance}}_{A B} = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$= \sqrt{{\left(19 - 2\right)}^{2} + {\left(31 - \left(- 3\right)\right)}^{2}}$

$= \sqrt{1445}$

$= 17 \sqrt{5}$