Question 406·Hard·Math

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For which value of $k$ will the equation $(x-2)^2=|x-k|$ have exactly three distinct real solutions?

A tangency is a one-point touch between graphs. When a parabola meets an absolute-value V, three solutions can come from one touch and two crossings; the V’s corner can also be shared by two branches. Complete the square to find an exact touching value, then graph both original sides in Desmos. Count distinct intersections, not the number of algebraic cases.

Hints

Hint 1
On the left of $k$ , $x-k$ is nonpositive, so $|x-k|=k-x$ . Rewrite that part of the equation to show which values of $k$ allow a left-side solution.
Hint 2
A completed square shows the smallest value a quadratic can have. What value of $k$ makes the left branch meet the parabola at one point?
Hint 3
A left-side touch gives only one solution. Graph the original two sides at that value of $k$ to check whether the right side supplies two more distinct solutions.

Step-by-step

Find a touch, then count intersections
Step 1
Write the left-side equation
One way to get three distinct solutions is a single touch on one side and two crossings on the other. Start where $x\le k$ . There, $x-k\le0$ , so the absolute value reverses its sign:
$(x-2)^2=k-x.$
Add $x$ to both sides to see which $k$ allows a left-side solution:
$k=(x-2)^2+x.$
Step 2
Expand the expression for $k$
Expand the square:
$k=x^2-4x+4+x.$
Combine the $x$ terms:
$k=x^2-3x+4.$
Now this quadratic, an expression containing $x^2$ , can be rewritten to show when it has one left-side solution.
Step 3
Find the exact left-side touch
Half of $-3$ is $-\frac32$ . Add and subtract its square, $\frac94$ , so the value stays the same:
$k=\left(x^2-3x+\frac94\right)+4-\frac94.$
Rewrite the first three terms as a square:
$k=\left(x-\frac32\right)^2+4-\frac94.$
A square is at least $0$ , and equals $0$ only at $x=\frac32$ . So the first left-side contact occurs at $k=4-\frac94$ , with one point. It belongs to this side because $\frac32\le4-\frac94$ .
Step 4
Check the other two intersections
Type $k=4-\frac94$ , $y=(x-2)^2$ , and $y=|x-k|$ on separate Desmos lines. Desmos shows $k=1.75$ , which is $\frac74$ . The graphs touch on the left at $x=1.5$ and cross twice on the right, at approximately $x=1.793$ and $x=3.207$ . These are three distinct intersections, with the touch counted once. So $k=\frac74$ gives exactly three distinct real solutions. Choice C.

Question 406·Hard·Math

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