Given the Quadratic Equation:
[tex]x^2-5x+6=0[/tex]1. Find the x-intercepts:
- Factor the equation by finding two numbers whose sum is -5 and whose product is 6. These are -2 and -3, because:
[tex]\begin{gathered} -2-3=-5 \\ (-2)(-3)=6 \end{gathered}[/tex]Then:
[tex](x-2)(x-3)=0[/tex]- Now you know that the x-intercepts are:
[tex]\begin{gathered} x_1=2 \\ x_2=3 \end{gathered}[/tex]2. Now you need to graph this function:
[tex]y=x^2-5x+6[/tex]In order to find the graph with better precision, you can find the vertex:
- Find the x-coordinate with this formula:
[tex]x=-\frac{b}{2a}[/tex]In this case, knowing that the function has the form:
[tex]y=ax^2+bx+c[/tex]You can identify that:
[tex]\begin{gathered} b=-5 \\ a=1 \end{gathered}[/tex]Then, you get:
[tex]x=-\frac{(-5)}{2(1)}=\frac{5}{2}=2.5[/tex]- Find the y-coordinate of the vertex by substituting the x-coordinate into the function and evaluating:
[tex]y=(2.5)^2-5(2.5)+6=-0.25[/tex]Hence, the vertex of the parabola is:
[tex](2.5,-0.25)[/tex]3. To find two other points on the parabola, you can substitute these values into the function and evaluate:
[tex]\begin{gathered} x=2.2 \\ x=2.7 \end{gathered}[/tex]Then, you get:
[tex]\text{For }x=2.2\rightarrow y=(2.2)^2-5(2.2)+6=-0.16[/tex][tex]\text{For }x=2.7\rightarrow y=(2.7)^2-5(2.7)+6=-0.21[/tex]Therefore, you know these two other points:
[tex]\mleft(2.2,-0.16\mright),\mleft(2.7,-0.21\mright)[/tex]Hence, the answer is:
