Chapter 1.9, Problem 1E

To determine

To fill: The blank in the statement “the solution of the equation $x^2-2x-3=0$ are the $$ intercepts of the graph $y=x^2-2x-3$ ”.

Answer to Problem 1E

The complete statement is the solution of the equation $x^2-2x-3=0$ are the $x$intercepts of the graph $y=x^2-2x-3$ .

Explanation of Solution

Given information:

The solution of the statement $x^2-2x-3=0$ are the $$ intercepts of the graph $y=x^2-2x-3$ .

Consider the provided statement “ $x^2-2x-3$ are the $$ intercepts of the graph $y=x^2-2x-3$ .” $x$ -intercepts is a point where a line crosses the x- axis.

  $y-$ intercepts is a point where a line crosses the $y-$ axis.

So, the equation is $x^2-2x-3=0$ , we check the equation for $y=0$ .

Here $y=0$ then the equation becomes:

  $\begin{array}{c}{x}^2-2x-3=0\\ x^2-2x-x-3=0\\ x\left(x-2\right)+1\left(x-2\right)=0\\ \left(x+1\right)\left(x-2\right)=0\end{array}$

Either $x+1=0$ or $x-2=0$ :

Simplified further as $x=-1$ or $x=2$ .

Substitute the value of $x$ in the equation $x^2-2x-3=y$ .

Now for $x=-1$ substitute the value of $x$ in a given equation $y=x^2-2x-3$ .

  $\begin{array}{c}y=x^2-2x-3\\ y=-1^2-2\cdot -1-3\\ y=1-2-3\\ y=-4\end{array}$

Now for $x=2$ substitute the value of $x$ in the given equation $y=x^2-2x-3$ .

  $\begin{array}{c}y=x^2-2x-3\\ y=2^2-2\cdot 2-3\\ y=4-4-3\\ y=-3\end{array}$

The graph of the equation is open upward.

  

Here the graph of line crosses the $x$ -axis ,so the graph statement $x^2-2x-3=0$ has $x$ intercepts, because there is no line of intersection at $y$ -axis.

Thus, the complete statement, “the solution of the statement $x^2-2x-3=0$ are the $x$ intercepts of the graph $y=x^2-2x-3$ ”.