After you set $x^2 + 2x - 1$ equal to $-x + 5$ and move all terms to one side, you should get a quadratic equation. Write it clearly in the form $ax^2 + bx + c = 0$ and identify $a$, $b$, and $c$.

Instead of solving the quadratic completely, recall: for $ax^2 + bx + c = 0$, how can you find the product of the two solutions using only $a$ and $c$?

Use your identified values of $a$ and $c$ from the quadratic you formed to compute $c/a$. That value is the product of the $x$-coordinates of the intersection points.

In Desmos, enter the two equations as

• y = x^2 + 2x - 1
• y = -x + 5

You will see a parabola and a line that intersect at two points.

Click or tap on each intersection point where the line and the parabola cross. Desmos will display the coordinates $(x_1, y_1)$ and $(x_2, y_2)$; note the two $x$-values.

In a new expression line, type the product of the two $x$-values you just read (for example, (first_x_value)*(second_x_value)). The resulting output is the product of the $x$-coordinates of the intersection points.

At intersection points, the $y$-values of the two graphs are equal, so set the right sides equal:

Move all terms to one side to get a quadratic equation:

The $x$-coordinates of the intersection points are the solutions of $x^2 + 3x - 6 = 0$.

You are asked for the product of the two $x$-coordinates, not the coordinates themselves.

For a quadratic of the form

with solutions (roots) $r_1$ and $r_2$, there is a useful relationship:

• Sum of roots: $r_1 + r_2 = -\dfrac{b}{a}$
• Product of roots: $r_1 r_2 = \dfrac{c}{a}$

This is often called Vieta's formulas and lets you find the product directly from the coefficients.

Compare $x^2 + 3x - 6 = 0$ with $ax^2 + bx + c = 0$:

• $a = 1$
• $b = 3$
• $c = -6$

The product of the two roots (the $x$-coordinates of the intersection points) is

So, the product of the $x$-coordinates of the intersection points is $-6$.