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 2 - 2022 - Paper 3

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

$2x - 5 = 6x^2 - 25x - 8$

Rearranging the equation gives:

$0 = 6x^2 - 27x - 3$

Next, we can use the quadratic formula to solve for x, where the coefficients are $a = 6$, $b = -27$, and $c = -3$:

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

Calculating the discriminant:

$b^2 - 4ac = (-27)^2 - 4 \cdot 6 \cdot (-3) = 729 + 72 = 801$

Then substituting back into the quadratic formula:

$x = \frac{27 \pm \sqrt{801}}{12}$

We can simplify $\sqrt{801}$ to find its approximate value:

$\sqrt{801} \approx 28.3$

Thus,

$x_1 = \frac{27 + 28.3}{12} \approx 4.6$

$x_2 = \frac{27 - 28.3}{12} \approx -0.1$

Next, we substitute these x-values back into the equation of L to find the corresponding y-values: For $x_1 \approx 4.6$:

$y = 2(4.6) - 5 \approx 4.2$

For $x_2 \approx -0.1$:

$y = 2(-0.1) - 5 \approx -5.2$

So, the points of intersection are approximately:

$(4.6, 4.2) \text{ and } (-0.1, -5.2)$