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 3 - 2022 - Paper 3

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

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

Rearranging the equation:

$0 = 6x^{2} - 25x - 8 - 2x + 5$

This simplifies to:

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

Now, we can use the quadratic formula to find the values of x:

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

Where:

• $a = 6$
• $b = -27$
• $c = -3$

Calculating the discriminant:

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

Now substituting these values into the formula:

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

We approximate $\sqrt{801} \approx 28.3$, leading to two values of x:

1. $x \approx \frac{27 + 28.3}{12} \approx 4.6$
2. $x \approx \frac{27 - 28.3}{12} \approx -0.1$

Next, we substitute these x-values back into the equation of line L to find their corresponding y-values:

For $x \approx 4.6$:

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

So one point of intersection is approximately $(4.6, 4.2)$.

For $x \approx -0.1$:

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

So the other point of intersection is approximately $(-0.1, -5.2)$.

In conclusion, the coordinates of the points of intersection are:

• $(4.6, 4.2)$
• $(-0.1, -5.2)$.