Given: #(x-2)(x+3)=x-10#
#color(brown)("Consider just the left hand side (LHS)")#
#color(blue)((x-2))color(green)( (x+3) )#
Multiply everything in the right brackets by everything in the left.
#color(green)(color(blue)(x)(x+3) color(white)("ddd")color(blue)(-2)(x+3) ) larr# Notice the minus followed the 2
#color(green)(x^2+ubrace(3xcolor(white)("dddd")-2x)-6)#
#color(green)(x^2color(white)("dddddd")+xcolor(white)("ddd")-6)#
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#color(brown)("Putting it all back together")#
#color(white)("dd")"LHS"color(white)("ddddd")=color(white)("ddd")"RHS"#
#x^2+x-6color(white)("dd")=color(white)("dd")x-10#
As #x# is on both sides we can cancel them out.
#x^2+cancel(x)-6color(white)("dd")=color(white)("dd")cancel(x)-10#
Add 10 to both sides
#x^2+4=0#
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#color(brown)("Completing the square")# ( If they insist !!!)
Write as #x^2+0x+4=0#
#(x+0/2)^2+4=0#
#(x+0/2)^2=-4#
Square root both sides
#x+0/2=+-sqrt(-4)#
#x=+-sqrt(4xx(-1))#
#x=+-2i#
