Question

|x²+4x+2|=(5x+16)/3​

Answer 1

Answer:

x^2+4x+2=(5x+16)/3

3x^2+12x+6=5x+16

3x^2+12x-5x+6-16=0

3x^2+7x-10=0

3x^2+10x-3x-10=0

x(3x+10)-1(3x+10)=0

(x-1)(3x+10)=0

x-1=0,3x+10=0

x=1,3x=-10

x=1,x=-10/3

Answer 2

Complete Question: Find the total number of roots of the equation |x²+4x+2| = (5x+16)/3​.

Answer:

The total number of the roots are $x=1$, $x=-2$, $x=-10/3$ and $x=-11/3$.

Step-by-step explanation:

Recall the definition of modulus function,

$|x|=\left \{ {{x,\ x\ \geq\ 0} \atop {-x,\ x\ < \ 0}} \right.$

Consider the given equation as follows:

$|x^2+4x+2|=\frac{5x+16}{3}$

Case1. When $(x^2+4x+2) > 0$. Then,

$x^2+4x+2=\frac{5x+16}{3}$

Cross multiply the equation as follows:

⇒ $3(x^2+4x+2)=5x+16$

⇒ $3x^2+12x+6=5x+16$

Simplify the equation as follows:

⇒ $3x^2+12x-5x+6-16=0$

⇒ $3x^2+7x-10=0$

Using splitting method, solve the quadratic equation.

⇒ $3x^2-3x+10x-10=0$

⇒ $3x(x-1)+10(x-1)=0$

⇒ $(x-1)(3x+10)=0$

⇒ $x=1$, $x=-10/3$

Case2. When $(x^2+4x+2) < 0$. Then,

$-(x^2+4x+2)=\frac{5x+16}{3}$

Cross multiply the equation as follows:

⇒ $-3(x^2+4x+2)=5x+16$

⇒ $-3x^2-12x-6=5x+16$

Simplify the equation as follows:

⇒ $-3x^2-12x-5x-6-16=0$

⇒ $-3x^2-17x-22=0$

⇒ $3x^2+17x+22=0$

Using splitting method, solve the quadratic equation.

⇒ $3x^2+11x+6x+22=0$

⇒ $x(3x+11)+2(3x+11)=0$

⇒ $(x+2)(3x+11)=0$

⇒ $x=-2$, $x=-11/3$

The total number of the roots are $x=1$, $x=-2$, $x=-10/3$ and $x=-11/3$.

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