Chapter 9.5, Problem 49PPE

a)

To determine

To evaluate: solution of the equation $x^2-6x+4=0$ , and to write the solutions in the form of $p\pm \sqrt{q}$ .

Answer to Problem 49PPE

The solution is $3\pm \sqrt{5}$ .

Explanation of Solution

Given: Equation is given as $x^2-6x+4=0$ .

Taking the given equation,

  $x^2-6x+4=0$

Solving,

  $\begin{array}{c}{x}^2-6x+4=0\\ x^2-6x+9-9+4=0\\ {\left(x-3\right)}^2-5=0\\ x-3=\sqrt{5}\\ x=3\pm \sqrt{5}\end{array}$

Hence, the solution is $3\pm \sqrt{5}$ .

b)

To determine

To evaluate: the coordinates of the vertex of the graph of $x^2-6x+4=0$

using the formula

  $x=-\frac{b}{2a}$ .

Answer to Problem 49PPE

The coordinate of the vertex is (3, −5).

Explanation of Solution

Given: Equation is given as $x^2-6x+4=0$ .

Taking the given equation,

  $x^2-6x+4=0$

In the above equation, $a=1,b=-6,c=4$

Now, finding the vertex,

  $\begin{array}{c}x=-\frac{b}{2a}\\ =-\frac{-6}{2}\\ =3\end{array}$

Now finding y-coordinate,

  $\begin{array}{l}y=x^2-6x+4\\ y={\left(3\right)}^2-6\left(3\right)+4\\ y=9-18+4\\ y=-5\end{array}$

So, the coordinate of the vertex is (3, −5).

c)

To determine

To evaluate: the relationship between answers in part a) and part b).

Explanation of Solution

In part a), the solutions of the given quadratic equation is calculated, whereas in part b), the coordinates of the same equation is calculated.

In part a), the equation touches the x-axis at two points that is why it is called solution.

In part b), the equation has one −point that is called as vertex.