L is the straight line with equation $y = 2x - 5$ C is a graph with equation $y = 6x^2 - 25x - 8$ Using algebra, find the coordinates of the points of intersection of L and C - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3

To find the intersection points of the line L and the curve C, we need to set their equations equal to each other:

1. Start with the equations: $y = 2x - 5$
   $y = 6x^2 - 25x - 8$

2. Set the equations equal: $2x - 5 = 6x^2 - 25x - 8$

3. Rearranging gives: $0 = 6x^2 - 27x - 3$

4. To solve this quadratic equation, we can use the quadratic formula, where $a = 6$, $b = -27$, and $c = -3$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $x = \frac{27 \pm \sqrt{(-27)^2 - 4(6)(-3)}}{2(6)}$ $x = \frac{27 \pm \sqrt{729 + 72}}{12}$ $x = \frac{27 \pm \sqrt{801}}{12}$ $x = \frac{27 \pm 9\sqrt{89}}{12}$

5. The approximation for the roots will yield the values for $x$ as: $x_1 \approx 5.5, \quad x_2 \approx -0.25$

6. Next, substitute these $x$ values back into the equation of L to find their corresponding $y$ values:

   • For $x_1 = 5.5$:
     $y = 2(5.5) - 5 = 6$
     So one intersection point is $(5.5, 6)$.
   • For $x_2 = -0.25$:
     $y = 2(-0.25) - 5 = -5.5$
     So the other intersection point is $(-0.25, -5.5)$.

7. Therefore, the coordinates of the points of intersection are: $(5.5, 6)$ and $(-0.25, -5.5)$.