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Find range of $m$ such that the equation $|x^2-3x+2|=mx$ has 4 distinct real solutions $\alpha,\beta,\gamma,\delta$

To show how I got the wrong answers.

From $|x^2-3x+2|=mx$

I got the two case $x^2-3x+2=mx$ when $x>2 $ or $ x<1$

and $x^2-3x+2=-mx$ when $1<x<2$

also $m\neq0$ (because if $m=0$ , This will given only 2 answers not 4)

try to find the first two answers $x^2-3x+2=mx$ when $x>2 $ or $ x<1$

$x^2-(3+m)x+2=0$ when $x>2 $ or $ x<1$

Use quadratic formula will given $x= \frac{3+m\pm \sqrt{(3+m)^2-4\times2}}{2}$

$x$ will be real number and have two answers if $\sqrt{(3+m)^2-4\times2} > 0$

$m^2+6m+11>0$, got that $-3-2\sqrt{2}<m<-3+2\sqrt{2}$

and on the another case where $x^2-3x+2=-mx$ when $1<x<2$

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

Use quadratic formula will given $x= \frac{3-m\pm \sqrt{(3-m)^2-4\times2}}{2}$

$x$ will be real number and have the others two answers if $\sqrt{(3-m)^2-4\times2} > 0$

$\sqrt{(3-m)^2-4\times2} > 0$, got that $3-2\sqrt{2}<m<3+2\sqrt{2}$

So, I believe that the answers should be $(-3-2\sqrt{2}<m<-3+2\sqrt{2}) \cup (3-2\sqrt{2}<m<3+2\sqrt{2})$

However, the book's right answer is $0 < m < 3-2\sqrt{2}$

Please show me the method to obtain the right answers.

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  • $\begingroup$ It is very simple, $mx>0$ by definition, so you can have $m<0$ only when $x<0$ as well... So you need to pay attention to the couple $mx$ in all your answers. $\endgroup$
    – Martigan
    Commented Apr 30, 2015 at 15:25

1 Answer 1

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There is some positive value $m$ such that $y=mx$ is tangent to $y=-(x^2-3x+2)$.

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This value must make $0$ the discriminant of the equation $$x^2-3x+2=-mx$$

That is, $$m^2-6m+1=0$$

The least root of this equation is $$3-2\sqrt2$$

So $0<m< 3-2\sqrt 2$.

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