Question 184·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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In the $xy$ -plane, the graph of $y = x^3 - 6x$ intersects the graph of $y = x$ at three points. One of the intersection points is $(0,0)$ , and the other two are $(a,a)$ and $(-a,-a)$ . What is the value of $a^2$ ?

Your answer

A nonlinear system has two equations that share variables, with at least one curved graph. At an intersection, both equations give the same $y$ -value. Graph them in Desmos to locate the crossings, then substitute a point’s coordinates to get an exact result. Check the zero crossing before dividing by an unknown, and don’t stop at a coordinate if the question asks for its square.

Hints

Hint 1
An intersection lies on both graphs. Graph each equation in Desmos and click the two crossings away from $(0,0)$ . Their coordinates will be rounded, but they’ll help you see which points the question calls the other two.
Hint 2
At $(a,a)$ , both the $x$ -coordinate and the $y$ -coordinate are $a$ . Put $a$ in for both variables in the cubic equation. What equation must hold at that point?
Hint 3
The point $(a,a)$ is not $(0,0)$ , so $a$ isn’t $0$ . That lets you divide an equation by $a$ without losing the point you need. Keep the target $a^2$ in view.

Step-by-step

Graph the crossings, then solve exactly
Step 1
Locate the nonzero intersections
Type both given equations in Desmos and click the two nonzero intersections. Desmos labels them about $(2.646,2.646)$ and $(-2.646,-2.646)$ . An intersection lies on both graphs. The labels are rounded, so use the equations for the exact value.
Step 2
Use the coordinates of $(a,a)$
The point $(a,a)$ lies on the cubic graph, so putting $a$ in for both $x$ and $y$ must make its equation true:
$a=a^3-6a.$
Step 3
Put zero on one side
Subtract $a$ from both sides so the equation has zero on one side:
$0=a^3-7a.$
Step 4
Divide only after checking $a$
$(a,a)$ is one of the other two points, not $(0,0)$ , so $a\ne0$ . Check the zero case before dividing by a variable. Divide both sides by $a$ :
$0=a^2-7.$
Step 5
Find the requested square
Add $7$ to both sides: $7=a^2$ . So the value of $a^2$ is $7$ . Grid in 7.

Question 184·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this