We want to solve the given quadratic equation by graphing.
x^2+7x+14=0
To do so, we will graph the quadratic function represented by the left-hand side of the above equation. To draw the graph, we must start by identifying the values of a, b, and c.
y=x^2+7x+14 ⇔ y= 1x^2+ 7x+ 14
We can see that a= 1, b= 7, and c= 14. Now, we will follow four steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=-3.5.
Calculating the Vertex
To calculate the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b.
Vertex: ( - b/2 a, f( - b/2 a ) )
Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=-3.5. Thus, the x-coordinate of the vertex is also -3.5. To find the y-coordinate, we need to substitute -3.5 for x in our function.
We found the y-coordinate, and now we know that the vertex is (-3.5,1.75).
Identifying the y-intercept and its Reflection
The y-intercept of the graph of a quadratic function written in standard form is given by the value of c. Thus, the point where our graph intercepts the y-axis is (0, 14). Let's plot this point and its reflection across the axis of symmetry.
Connecting the Points
We can now draw the graph of the function. Since a= 1, which is positive, the parabola will open upward. Let's connect the three points with a smooth curve.
The x-intercepts of the graph are the solutions to the given equation. However, this graph has no x-intercepts. Therefore, this equation has no real solutions.
