The solutions, or roots, of ax^2+bx+c=0 are the x-intercepts of the graph of f(x)=ax^2+bx+c. Our equation is already written in standard form. Let's identify the function related to the equation.
Equation:& x^2+8x=0
Related Function:& f(x)=x^2+8x
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+8x ⇔ f(x)=1x^2+8x+
We can see that a=1, b=8, and c= . Now, we will follow three steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=- 4.
Making the Table of Values
Next, we will make a table of values using x values around the axis of symmetry x=- 4.
x
x^2+8x
f(x)
- 8
( - 8)^2+8( - 8)
0
- 6
( - 6)^2+8( - 6)
- 12
- 4
( - 4)^2+8( - 4)
- 16
- 2
( - 2)^2+8( - 2)
- 12
0
0^2+8( 0)
0
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 ( - 8,0) and ( 0,0). Therefore, the equation x^2+8x=0 has two solutions, x= - 8 and x= 0.
