Question 889·Medium·Math

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The function $q$ is defined by

q(x) = -2\bigl(x − 4\bigr)^2 + 18

Which of the following values of $x$ satisfies $q(x) = 2$ ?

A quadratic function can produce the same output at two different inputs. When a question asks which input gives a specified output, graph the function and a horizontal line at that output in Desmos, then click both intersections. Compare their x-coordinates with the answer choices by evaluating the radicals, rather than guessing from rounded graph labels.

Hints

Hint 1
An input is an x-value, and the output is $q(x)$ . To find an input that gives the requested output, set the whole function rule equal to that output. What horizontal line represents it on a graph?
Hint 2
An intersection is a point shared by two graphs. At each crossing of the function and the target line, the x-coordinate is a possible input. Check both crossings because a quadratic can reach one height twice.
Hint 3
Desmos labels intersections with rounded decimals, while the choices contain exact square roots. Evaluate each choice on its own Desmos line. Which decimal matches the x-coordinate of a crossing?

Step-by-step

Approach 1: Graph the target output

Step 1
Turn the requested output into an equation
The question asks for an input $x$ whose output $q(x)$ is $2$ , so set the entire rule equal to $2$ :
$-2(x-4)^2+18=2.$
Step 2
Find where the function reaches that output
Type $y=-2(x-4)^2+18$ and $y=2$ in Desmos. Click both intersections. Desmos shows approximately $(1.172,2)$ and $(6.828,2)$ . At an intersection, the x-coordinate is an input that gives the target output.
Step 3
Match an exact choice to an intersection
Type the four radical choices on separate Desmos lines. In order, they display about $2.586$ , $5.414$ , $8.243$ , and $1.172$ . Only $4-2\sqrt{2}$ matches an intersection's x-coordinate, so it satisfies $q(x)=2$ . Choice D.

Approach 2: Solve the squared equation exactly

Step 1
Move the constant
Instead of reading decimals from the graph, start with $-2(x-4)^2+18=2$ . Subtract $18$ from both sides to isolate the squared term:
$-2(x-4)^2=-16.$
Step 2
Remove the coefficient
Divide both sides by $-2$ ; keep the negative sign with the coefficient:
$(x-4)^2=8.$
Step 3
Keep both square roots
Take square roots of both sides. A square has both a positive and a negative root, so keep both signs:
$x-4=\pm\sqrt{8}.$
Step 4
Simplify the radical
Because $8=4\cdot2$ and $\sqrt{4}=2$ , simplify the radical:
$x-4=\pm2\sqrt{2}.$
Step 5
Find the listed input
Add $4$ to both sides: $x=4\pm2\sqrt{2}$ . The minus version is the listed value that gives $q(x)=2$ . Choice D.

Question 889·Medium·Math

Strategy

Hints

Step-by-step

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Strategy

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