A Level H2 Maths 2012 Paper 2 Q3 Solution

(i)

(The graph above is drawn using the Geogebra software 🙂 )

(ii)

$x^3+x^2-2x-4=4$

$x^3+x^2-2x-8=0$

By GC, $x=2$

By long division, $x^3+x^2-2x-8=(x-2)(x^2+3x+4)$

The discriminant of $x^2+3x+4$ is

$D=b^2-4ac=3^2-4(1)(4)=-7<0$

Hence, there are no other real solutions (proven).

(iii) $x+3=2$

$x=-1$

(iv)

(v)

$|x^3+x^2-2x-4|=4$

$x^3+x^2-2x-4=4$ or $x^3+x^2-2x-4=-4$

$x^3+x^2-2x-8=0$ or $x^3+x^2-2x=0$

$x^3+x^2-2x-8=0 \implies x=2$ (from part ii)

$x^3+x^2-2x=x(x^2+x-2)=x(x-1)(x+2)=0$

$x=0,1,-2$

In summary, the roots are $-2,0,1,2$

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