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.
12x^2=- 11x+15
⇕
12x^2+11x-15=0
Now we can identify the function related to the equation.
Equation:& 12x^2+11x-15=0
Related Function:& f(x)=12x^2+11x-15
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)=12x^2+11x-15
⇕
f(x)= 12x^2+ 11x+( - 15)
We can see that a= 12, b= 11, and c= - 15. Now, we will follow three steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=- 1124.
Making the Table of Values
Next, we will make a table of values using x values around the axis of symmetry x=- 1124.
x
x^2-5x
f(x)
- 5/3
12( - 5/3)^2+11( -5/3)-15
0
- 11/12
12( - 11/12)^2+11( -11/12)-15
- 15
- 11/24
12( - 11/24)^2+11( -11/24)-15
≈ - 17.5208
0
12( 0)^2+11( 0)-15
- 15
3/4
12( 3/4)^2+11( 3/4)-15
0
Plotting and Connecting the Points
We can finally draw the graph of the function. Since a= 12, 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 ( - 53,0) and ( 34,0). Therefore, the equation 12x^2=- 11x+15 has two solutions, x= - 53 and x= 34.
