Excellence Part

L2MAEL

---

For ‘Excellence’, an ‘inequality’ using discriminants must be found (+ linking the context).

For example, when you have the following two functions:

$$y=\frac{1}{x-2}+3$$

$y=-kx+4$ and you’re asked to find the suitable range of values for $k$, so that the linear line doesn’t touch the hyperbola. This is done using discriminants.

Referring back to Discriminants

In Algebra, we used discriminants to find how many ‘solutions’, or ‘intersections’ along the $x$ axis there are for a function. For an equation like:

$$a{x}^2+bx+c=0$$

we can also say that the equation is equal to $y$:

$$y=a{x}^2+bx+c=0$$

Solving (factorizing) that quadratic function returns the values for $x$, where $y=0$. The discriminant of that quadratic function finds how many times the function intersects where $y=0$ (which is the $x$-axis).

Now, going back to Systems of Equations, when we try to find the intersections, we make the two functions equal the same:

$$\frac{1}{x-2}+3=-kx+4$$

Then we remove the denominator:

$$1+3(x-2)=(-kx+4)(x-2)$$ $$1+3x-6=-k{x}^2+2kx+4x-8$$

Reorder:

$$-k{x}^2+2kx+x-3=0$$

Grouping like terms:

$$-k{x}^2+(2k+1)x-3=0$$

Now, the discriminant of the combined equation indicates how many times the functions intersect each other (hyperbola and linear).

$$\Delta =(2k+1{)}^2-4(-k)(-3)$$

Now, we must give the discriminant a value in order to calculate $k$.

Since the question is asking us to find the values for $k$ so that there are no intersections among the two functions, the discriminant must be below zero:

$$(2k+1{)}^2-4(-k)(-3)<0$$ $$4k^2+4k+1-12k<0$$ $$4k^2-8k+1<0$$ $$k\approx 0.134\text{ or }1.866$$

$k$ would be between the values:

$$0.134<k<1.866$$

for there to be no intersections.

Now, for Excellence, refer it back to the context.