We are asked to solve the given quadratic equation. We will solve it by graphing. To solve a quadratic equation by graphing, we have to follow three steps.
The solutions, or roots, of ax^2+bx+c=0 are the x-intercepts of the graph of f(x)=ax^2+bx+c.
Now we can identify the function related to the equation.
Equation:& - x^2-5x+1=0
Related Function:& f(x)=- x^2-5x+1
Graphing the Related Function
To draw the graph of the related quadratic function written in standard form, we must start by identifying the values of a, b, and c.
f(x)=- x^2-5x+1
⇕
f(x)=( - 1)x^2+( - 5)x+1
We can see that a= - 1, b= - 5, and c= 1. 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+1
f(x)
- 7.5
-( - 7.5)^2-5( - 7.5)+1
- 17.75
- 5
-( - 5)^2-5( - 5)+1
1
- 2.5
-( - 2.5)^2-5( - 2.5)+1
7.25
0
- 0^2-5( 0)+1
1
2.5
- 2.5^2-5( 2.5)+1
- 17.75
Plotting and Connecting the Points
We can finally draw the graph of the function. Since a=- 1, which is negative, the parabola will open downwards. 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 ( - 5.2,0) and ( 0.2,0). Therefore, the equation - x^2-5x+1=0 has two solutions, approximately x= - 5.2 and x= 0.2.
