Exercise 12
1. In the diagram, the graphs of $y=x^{2}-2 x-3$, $y=-2$ and $y=x$ have been drawn. Use the graphs to find approximate solutions to the following equations:
a) $x^{2}-2 x-3=-2$
b) $x^{2}-2 x-3=x$
c) $x^{2}-2 x-3=0$
d) $x-2 x-1=0$

Step 1 ：a) To solve the equation $x^2 - 2x - 3 = -2$, we can rewrite it as $x^2 - 2x - 1 = 0$

Step 2 ：b) To solve the equation $x^2 - 2x - 3 = x$, we can rewrite it as $x^2 - 3x - 3 = 0$

Step 3 ：c) To solve the equation $x^2 - 2x - 3 = 0$, we can use the given equation directly

Step 4 ：d) To solve the equation $x - 2x - 1 = 0$, we can rewrite it as $-x - 1 = 0$ or $x = -1$

Step 5 ：Now, we can use the graphs to find approximate solutions for each equation:

Step 6 ：a) For $x^2 - 2x - 1 = 0$, the graph of $y = x^2 - 2x - 1$ intersects the line $y = -2$ at approximately $x \approx 1$ and $x \approx 3$

Step 7 ：b) For $x^2 - 3x - 3 = 0$, the graph of $y = x^2 - 2x - 3$ intersects the line $y = x$ at approximately $x \approx -1$ and $x \approx 3$

Step 8 ：c) For $x^2 - 2x - 3 = 0$, the graph of $y = x^2 - 2x - 3$ intersects the $x$-axis at approximately $x \approx -1$ and $x \approx 3$

Step 9 ：d) For $-x - 1 = 0$, the graph of $y = x - 2x - 1$ intersects the $x$-axis at $x = \boxed{-1}$