Chapter 9.5, Problem 54E

a.

To determine

To solve the equation with graphing method.

Answer to Problem 54E

Graph:

  

Explanation of Solution

Given:

Equation:

  $3x^2+11x+6=0$

Calculation:

On comparing the equation with $a{x}^2+bx+c$

  $a=3,b=11,c=6$

Step:1 Finding the axis of symmetry.

  $x=\frac{-b}{2a}$

So,

  $x=\frac{-11}{2(3)}$

Therefore,

  $x=\frac{-11}{6}$

So,

  $x=-1.83$

Step:2 $a=1$ , therefore $a>0$ so graph will open up.

Step:3

Find the plotting points:

  $y=x^2+4x+4$

For $x=1.83$ find $y$

  $y=3{(-1.83)}^2+11(-1.83)+6$

Opening brackets,

  $y=3(3.34)-20.13+6$

Again

  $y=10.02-14.13$

So,

  $y=-4.083$

Again, for $x=-1$ find $y$

  $y=3{(-1)}^2+11(-1)+6$

So,

  $y=3-11+6$

So,

  $y=-2$

Again, for $x=1$ find $y$

  $y=3{(1)}^2+11(1)+6$

So,

  $y=3+11+6$

So,

  $y=20$

Therefore plotting points are:

x	-1	-1.83	1
y	-2	-4.083	20

Plot the graph:

Plot all the points and axis of symmetry.

  

b.

To determine

To solve the equation with factoring method.

Answer to Problem 54E

  $x=-3$ or $x=\frac{-2}{3}$

Explanation of Solution

Given:

Equation:

  $3x^2+11x+6=0$

Calculation:

Split the middle term in such a way that the sum comes 11 and product comes 18

  $3x^2+2x+9x+6=0$

Taking common

  $x(3x+2)+3(3x+2)=0$

So,

  $(x+3)(3x+2)=0$

Therefore

  $(x+3)=0$ or $(3x+2)=0$

So,

  $x=-3$ or $3x=-2$

So,

  $x=-3$ or $x=\frac{-2}{3}$

c.

To determine

To solve the equation with quadratic formula.

Answer to Problem 54E

  $x=-3$ or $x=\frac{-2}{3}$

Explanation of Solution

Given:

Equation:

  $3x^2+11x+6=0$

Calculation:

On comparing the equation with $a{x}^2+bx+c$

  $a=3,b=11,c=6$

Finding the value of $x$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Substituting the values,

  $x=\frac{-(11)\pm \sqrt{{(11)}^2-4(3)(6)}}{2(3)}$

Opening brackets,

  $x=\frac{-11\pm \sqrt{121-72}}{6}$

So,

  $x=\frac{-11\pm \sqrt{49}}{6}$

Solving root,

  $x=\frac{-11\pm 7}{6}$

Therefore

  $x=\frac{-11+7}{6}$ or $x=\frac{-11-7}{6}$

So,

  $x=\frac{-4}{6}$ or $x=\frac{-18}{6}$

So,

  $x=-3$ or $x=\frac{-2}{3}$