The x-coordinates of the points of intersection between the line $y=2 x+3$ and the curve $y=x^2-x-1$ are (smaller first):
$x=$ □ and $x=$ □
To find the points where the line \(y = 2x + 3\) and the curve \(y = x^2 - x - 1\) intersect, we must set their y-values equal to each other. This is because at any point of intersection, the coordinates \((x, y)\) must satisfy both equations.
Setting the equations equal, we have:
\[2x + 3 = x^2 - x - 1\]
Now, we rearrange the equation to put it into the standard quadratic form, \(ax^2 + bx + c = 0\). Subtracting \(2x\) and \(3\) from both sides of the equation yields:
\[0 = x^2 - x - 2x - 1 - 3\]
\[x^2 - 3x - 4 = 0\]
We look for two numbers that multiply to \(-4\) and add to \(-3\). These numbers are \(-4\) and \(1\). Thus, we can factor the quadratic equation as:
\[(x - 4)(x + 1) = 0\]
The equation \((x - 4)(x + 1) = 0\) is satisfied when either factor equals zero.
Setting each factor to zero, we get:
Comparing the two values, \(-1\) is the smaller root and \(4\) is the larger root. Thus, the x-coordinates of the intersection points are \(-1\) and \(4\).
Understanding intersection points is essential in many fields, such as economics and engineering. For example, in economics, the "equilibrium price" is found by determining where the supply curve and the demand curve intersect. If supply is given by a function \(S(p)\) and demand by \(D(p)\), solving \(S(p) = D(p)\) allows a company or government to find the price at which the quantity demanded matches the quantity supplied, ensuring market stability. Similarly, in physics, finding where a trajectory curve intersects a target plane helps in calculating precisely when and where an object will land.