Solve 12(x-1)^2=f(1)

$$12 ( x - 1 ) ^ { 2 } = f ( 1 )$$

Solve for f

$$f=12\left(x-1\right)^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$12\left(x^{2}-2x+1\right)=f\times 1$$

Use the distributive property to multiply 12 by x^{2}-2x+1.

$$12x^{2}-24x+12=f\times 1$$

Swap sides so that all variable terms are on the left hand side.

$$f\times 1=12x^{2}-24x+12$$

Reorder the terms.

$$f=12x^{2}-24x+12$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{3f}}{6}+1$$

$$x=\frac{\sqrt{3f}}{6}+1$$

Solve for x

$$x=-\frac{\sqrt{3f}}{6}+1$$

$$x=\frac{\sqrt{3f}}{6}+1\text{, }f\geq 0$$