Mathematics - Quadratic Equation Question with Solution | TestHub

Given the equation $|x-a|=(x-1{)}^2$, we can deduce that $a=x\pm (x-1{)}^2$.

The number of solutions corresponds to the number of intersections between the horizontal line $y=a$ and the functions $f(x)=x^2-x+1$ and $g(x)=-x^2+3x-1$.

The graphs of $f(x)$ and $g(x)$ are tangent to each other at point $A(1,1)$.

The line $y=a$ intersects the two graphs at three points if and only if it passes through one of the three points $A$, $B$, or $C$.

Here, $B=\left(\frac{1}{2},\frac{3}{4}\right)$ is the vertex of $f(x)$, and $C=\left(\frac{3}{2},\frac{5}{4}\right)$ is the vertex of $g(x)$.

Therefore, the possible values for $a$ are $a\in \left\{\frac{3}{4},\frac{5}{4},1\right\}$.