We are asked to solve the given by graphing. There are three steps to solving a quadratic equation by graphing.
- Write the equation in , ax^2+bx+c=0.
- Graph the related f(x)=ax^2+bx+c.
- Find the , if any.
The solutions, or roots, of ax^2+bx+c=0 are the x-intercepts of the graph of f(x)=ax^2+bx+c. Let's write our equation in standard form. This means gathering all of the terms on the left-hand side of the equation.
x^2+8=-6x⇔ x^2+6x+8=0
Now we can identify the function related to the equation.
Equation:& x^2+6x+8=0
Related Function:& f(x)=x^2+6x+8
Graphing the Related Function
To draw the graph of the related function written in , we must start by identifying the values of a, b, and c.
f(x)=x^2 + 6x + 8⇔ f(x)= 1x^2 + 6x + 8
We can see that a= 1, b= 6, and c= 8. Now, we will follow three steps to graph the function.
- Find the .
- Make a using x values around the axis of symmetry.
- Plot and connect the points with a .
Finding the Axis of Symmetry
The axis of symmetry is a with equation x=- b2 a. Since we already know the values of a and b, we can substitute them into the formula.
The axis of symmetry of the parabola is the vertical line with equation x=-3.
Making the Table of Values
Next, we will make a table of values using x-values around the axis of symmetry x=-3.
| x
|
x^2+6x+8
|
f(x)
|
| -5
|
( -5)^2+6( -5)+8
|
3
|
| -4
|
( -4)^2+6( -4)+8
|
0
|
| -3
|
( -3)^2+6( -3)+8
|
-1
|
| -2
|
( -2)^2+6( -2)+8
|
0
|
| -1
|
( -1)^2+6( -1)+8
|
3
|
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 ( -4,0) and ( -2,0). Therefore, the equation x^2+8=-6x has two solutions, x= -4 and x= -2.
