Chapter 2.5, Problem 20E

To determine

Find the zeros of the function $g(x)=x^2+10x+23$ and write the polynomial as a product of linear factors and verify zeros using graphical method.

Answer to Problem 20E

  $-5+\sqrt{2},-5-\sqrt{2}$

Explanation of Solution

Given:

Function: $g(x)=x^2+10x+23$

Formula used:

The zeros of a quadratic equation $a{x}^2+bx+c=0$ can be found using the formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

  $\Rightarrow g(x)=x^2+10x+23$

To find zeros, put $g(x)=0,$

  $\Rightarrow x^2+10x+23=0$

Applying above formula,

  $\begin{array}{l}\Rightarrow x=\frac{-10\pm \sqrt{{(10)}^2-4(1)(23)}}{2(1)}\\ \Rightarrow x=\frac{-10\pm \sqrt{100-92}}{2}\\ \Rightarrow x=\frac{-10\pm \sqrt{8}}{2}\\ \Rightarrow x=\frac{-10\pm 2\sqrt{2}}{2}\\ \Rightarrow x=-5\pm \sqrt{2}\\ \Rightarrow x=-5+\sqrt{2},-5-\sqrt{2}\end{array}$

Now, writing the given function as product of linear factors,

  $\Rightarrow g(x)=(x-(-5+\sqrt{2}))(x-(-5-\sqrt{2}))$

Calculation for graph:

Consider $g(x)=x^2+10x+23$

Values of x	Values of f (x)
0	23
1	34
-1	14
2	47
-2	7

By taking different values of x, the graph can be plotted.

Graph:

  

Interpretation:

By observing graph, it is clear that the curve of the function meets x-axis at

  $x=-5+\sqrt{2}\text{and}-5-\sqrt{2}.$

Hence, the zeros of the function are $x=-5+\sqrt{2}\text{and}-5-\sqrt{2}.$

Conclusion:

Therefore, the zeros of given function are $x=-5+\sqrt{2}\text{and}-5-\sqrt{2}.$