The graph of $y = 2x^2 + 3x - 9$ is drawn below - OCR - GCSE Maths - Question 19 - 2020 - Paper 6

To find the values of $a$ and $b$, we need to determine the slope and y-intercept of the line $y = ax + b$ that intersects the curve $y = 2x^2 + 3x - 9$ at a specific point. Observing the graph, choose a point of intersection, for example, $(-1, -8)$. The slope of the line $a$ can be calculated as follows:

1. Calculate the change in $y$ over the change in $x$ using another point on the curve, for example, $(0, -9)$: $a = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-9 - (-8)}{0 - (-1)} = \frac{-1}{1} = -1$
2. The y-intercept $b$ can be determined from the equation $y = ax + b$: $-8 = -1(-1) + b \Rightarrow b = -8 + 1 = -7$ Thus, $a = -1$ and $b = -7$.

To solve the equation $2x^2 + x - 4 = 0$ using the graph, we will plot the line $y = -1x - 7$. The $y$ value of this line at the points where it intersects the curve $y = 2x^2 + 3x - 9$ helps us find the solutions: Observe where the line intersects the curve. From the graph, the intersection points occur at approximately:

$x = -2$ and $x = 3$. Thus, the solutions are:

$x = -2$ or $x = 3$.