The solutions, or roots, of ax^2+bx+c=0 are the x-intercepts of the graph of f(x)=ax^2+bx+c. Notice that our equation is already in standard form. Now we can identify the function related to the equation.
Equation:& x^2+5x+6=0
Related Function:& f(x)=x^2+5x+6
Graphing the Related Function
To draw the graph of the related function written in standard form, we must start by identifying the values of a, b, and c.
f(x)=x^2 + 5x + 6 ⇔ f(x)= 1x^2 + 5x + 6
We can see that a= 1, b= 5, and c= 6. Now, we will follow three steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=-2.5.
Making the Table of Values
Next, we will make a table of values using x values around the axis of symmetry x=-2.5.
x
x^2+5x+6
f(x)
-4
( -4)^2+5( -4)+6
2
-3
( -3)^2-5( -3)+6
0
-2.5
( -2.5)^2+5( -2.5)+6
- 0.25
-2
( -2)^2+5( -2)+6
0
-1
( -1)^2+5( -1)+6
2
Plotting and Connecting the Points
We can finally draw the graph of the function. Since a= 1, which is positive, the parabola will open upwards. Let's connect the points with a smooth curve.
Finding the x-intercepts
Let's identify the x-intercepts of the graph of the related function.
We can see that the parabola intersects the x-axis twice. The points of intersection are ( -3,0) and ( -2,0). Therefore, the equation x^2+5x+6=0 has two solutions, x= -3 and x= -2.
