Find the intersection points of the line y = -4x + 2.5 and the curve y = 2.44x/(1 + 1.44x).

Step-by-step solution

1. 1

   Set the two expressions for y equal. At an intersection both formulas return the same $y$, so $-4x+2.5 = \frac{2.44x}{1+1.44x}.$ Note the pole at $1+1.44x=0$, i.e. $x=-\tfrac{1}{1.44}=-0.6944\ldots$, which must be excluded from any answer.

2. 2

   Clear the denominator. Multiplying both sides by $1+1.44x$ (legal away from the pole) gives $(-4x+2.5)(1+1.44x) = 2.44x$. Expanding the left side: $-4x-5.76x^2+2.5+3.6x = -5.76x^2-0.4x+2.5$.

3. 3

   Collect into standard quadratic form. Moving $2.44x$ across, $-5.76x^2-2.84x+2.5=0$; multiplying by $-1$ and then by $100$ and halving gives the integer form $288x^2+142x-125=0.$ Integer coefficients make the discriminant exact.

4. 4

   Solve the quadratic. $\Delta = 142^2+4(288)(125) = 20164+144000 = 164164 = 4\cdot 41041$, so $x = \frac{-142 \pm 2\sqrt{41041}}{576} = \frac{-71 \pm \sqrt{41041}}{288}.$ With $\sqrt{41041}\approx 202.58578$, $x_1 \approx 0.456895$ and $x_2 \approx -0.949951$. Neither equals $-0.6944$, so both survive the pole check.

5. 5

   Get the y-values from the line. The line is the cheaper substitution: $y_1 = -4(0.456895)+2.5 \approx 0.672420$ and $y_2 = -4(-0.949951)+2.5 \approx 6.299803$.

6. 6

   Verify on the curve, not just the line. $\dfrac{2.44(0.456895)}{1+1.44(0.456895)} \approx 0.672420$ and $\dfrac{2.44(-0.949951)}{1+1.44(-0.949951)} \approx 6.299803$ - both match, so the two points really lie on both graphs. The second point sits on the far branch of the hyperbola, past the vertical asymptote.