Question 117·200 Super-Hard SAT Math Questions·Advanced Math

0 / 200 Questions completed

In the given system of equations, $a$ is a constant. The system has two distinct real solutions, and in both solutions the $x$ -coordinate is negative. Which choice could be the value of $a$ ?

y = x + 1

y = a(x - 2)^2

A line and a parabola form a linear-quadratic system. Each point where they intersect is a solution. When the choices give values for a constant, graph the equations in Desmos and test each value. Check both conditions: two separate intersections and the required sign of each $x$ -coordinate. A parabola opening downward does not guarantee two intersections.

Hints

Hint 1
An intersection is a point on both graphs, so it satisfies both equations. A line and a parabola can meet zero, one, or two times. How many separate meeting points do you need here?
Hint 2
A square such as $(x-2)^2$ cannot be negative. If $a<0$ , compare the parabola’s height with $x+1$ when $x\ge0$ . Could the graphs meet at a nonnegative $x$ ?
Hint 3
A tangent line touches a parabola at one point. If the graphs appear to touch as you change $a$ in Desmos, count that point once, not twice.

Step-by-step

Test each value on the graph
Step 1
Check whether two intersections are enough
Type $a=\frac14$ , $y=x+1$ , and $y=a(x-2)^2$ . Click both intersections, where the graphs share a point. Desmos shows $(0,1)$ and $(8,9)$ . There are two solutions, but neither $x$ -coordinate is negative. In particular, $0$ is not negative.
Step 2
Locate intersections when $a$ is negative
A square is never negative, so if $a<0$ , the parabola has $a(x-2)^2\le0$ . But if $x\ge0$ , the line has $x+1\ge1$ . Those heights cannot be equal. So any intersections for negative $a$ must have negative $x$ . You still need to check that there are two.
Step 3
Test the first negative value
Change the Desmos value to $a=-\frac18$ . The parabola and line do not intersect, so this value gives no real solutions, even though its intersections, if any, would have negative $x$ .
Step 4
Distinguish a touch from two crossings
Change $a$ to $-\frac{1}{12}$ . Desmos shows the line touching the parabola at $(-4,-3)$ . This tangent point is one intersection, not two distinct solutions.
Step 5
Find two negative $x$ -coordinates
Change $a$ to $-\frac{1}{24}$ and click both intersections. Desmos shows about $(-18.485,-17.485)$ and $(-1.515,-0.515)$ . The first number in each ordered pair is $x$ , so there are two distinct intersections, both with negative $x$ -coordinates. The value of $a$ could be $-\frac{1}{24}$ . Choice C.

Question 117·200 Super-Hard SAT Math Questions·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this