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

We can use the quadratic formula to solve for x:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a = 6$, $b = -27$, and $c = -3$. Calculating the discriminant:

$$D = (-27)^2 - 4(6)(-3) = 729 + 72 = 801$$

Thus, we find:

$$x = \frac{27 \pm \sqrt{801}}{2(6)} = \frac{27 \pm 3\sqrt{89}}{12}$$

Using the x-values found earlier, we substitute them back into the equation of line L to find the corresponding y-values: For:

$$x_1 = \frac{27 + 3\sqrt{89}}{12}$$ $$y_1 = 2\left( \frac{27 + 3\sqrt{89}}{12} \right) - 5$$

And for:

$$x_2 = \frac{27 - 3\sqrt{89}}{12}$$ $$y_2 = 2\left( \frac{27 - 3\sqrt{89}}{12} \right) - 5$$

Thus, the points of intersection are:

$$\left( \frac{27 + 3\sqrt{89}}{12} , y_1 \right) \text{ and } \left( \frac{27 - 3\sqrt{89}}{12} , y_2 \right)$$