Question

Number of integral solutions of the inequation $x^2-10x+25sgn(x^2+4x-32)\le 0$

• A. $infinite$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is D $8$

We nave the equation........................

$x^2-10x+25sgn(x^2+4x-32)\le 0$

first we solve,

$\begin{array}{c}{x}^2-10x+25sgn(x^2+4x-32)\le 0\\ \\ \Rightarrow x^2+4x-32=(x+8)(x-4)\\ if,\\ sgn(x^2+4x-32)=1x<-8orx>4\\ 0x=4,-8\\ -1x\in (-8,4)\\ letsthevalueoffunctionis1,\\ sgn\left(f\right(x)=1\\ \Rightarrow x^2-10x+25\le 0\\ \Rightarrow (x-5{)}^2\le 0,\therefore x=5\\ and,sgn\left(f\right(x)=0\\ \Rightarrow x^2-10x+25\le 0\\ \Rightarrow x^2-10x\le 0\\ Again,\\ if,valueofx\in \left[0,10\right]\\ \therefore x=4\\ Again,sgn\left(f\right(x)=-1\\ eq{u}^{n}is,x^2-10x-25\le 0\\ \Rightarrow x=\frac{10\pm \sqrt{100+100}}{2}\\ \Rightarrow =5+5\sqrt{2}\\ thatmeans,thevalueofx\in \left[5-5\sqrt{2},5+5\sqrt{2}\right]\\ \left[-\mathrm{2.1........................12.1}\right]\\ Now,integeris\left[-\underset{–}{2,-1,0,1,2,3,}4,5,............13\right]\\ so,thattotalno.ofinteger(x=8),thecorrectoptionisD.\end{array}$