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
