The number of real roots of the equation x|x- | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) We analyze the equation $x|x-2|+3|x-3|+1=0$ in three cases:Case $(I): x < 2$The equation becomes $x(2-x) + 3(3-x) + 1 = 0$$-x^2 + 2x + 9 - 3x + 1

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(C) We analyze the equation $x|x-2|+3|x-3|+1=0$ in three cases:Case $(I): x The equation becomes $x(2-x) + 3(3-x) + 1 = 0$$-x^2 + 2x + 9 - 3x + 1 = 0$$-x^2 - x + 10 = 0 \Rightarrow x^2 + x - 10 = 0$The roots are $x = \frac{-1 \pm \sqrt{1 - 4(1)(-10)}}{2} = \frac{-1 \pm \sqrt{41}}{2}$.Since $\sqrt{41} \approx 6.4$,$x_1 = \frac{5.4}{2} = 2.7$ (not in range) and $x_2 = \frac{-7.4}{2} = -3.7$ (in range).So,$1$ root here.Case $(II): 2 \leq x The equation becomes $x(x-2) + 3(3-x) + 1 = 0$$x^2 - 2x + 9 - 3x + 1 = 0$$x^2 - 5x + 10 = 0$The discriminant $D = (-5)^2 - 4(1)(10) = 25 - 40 = -15 No real roots in this interval.Case $(III): x \geq 3$The equation becomes $x(x-2) + 3(x-3) + 1 = 0$$x^2 - 2x + 3x - 9 + 1 = 0$$x^2 + x - 8 = 0$The roots are $x = \frac{-1 \pm \sqrt{1 - 4(1)(-8)}}{2} = \frac{-1 \pm \sqrt{33}}{2}$.Since $\sqrt{33} \approx 5.74$,$x_3 = \frac{4.74}{2} = 2.37$ (not in range) and $x_4 = \frac{-6.74}{2} = -3.37$ (not in range).No real roots in this interval.Thus,there is only $1$ real root.

The number of real roots of the equation $x|x-2|+3|x-3|+1=0$ is:

Similar Questions

If ${x^2} + 2x + 2xy + my - 3$ has two rational factors,then the value of $m$ will be

The number of pairs of consecutive positive even integers such that the sum of their squares is $290$ is

If $\frac{1+\sqrt{3}i}{2}$ is a root of the equation $x^4-x^2+x-1=0$,then its real roots are:

Which among the following equations has roots that are negatives of the roots of the equation $x^3-x^2+x-4=0$?

The number of distinct quadratic equations $ax^2 + bx + c = 0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c$ such that $a \neq b \neq c$ from the set $\{0, 1, 2, 4\}$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module