In a quadratic equation, the value of the related quadratic function is 0. For example, if you substitute 0 for y in the quadratic function, the result is the quadratic equation 0 = ax2 + by + c. The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by finding the x-intercepts or zeros of the related quadratic function.
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1. The Buckingham Fountain in Chicago has 133 jets through which water
flows. The path of water streaming from a jet is in the shape of a parabola.
Find the distance from the jet where the water hits the ground by graphing. Use
the function h(d) = –2d2
+ 4d + 6, where h(d) represents the height of a stream of
water at any distance d from its jet in feet.
Alternative Solutions:
2.
Find the roots of x2 – 10x +
16 = 0 by graphing the related function.
Alternative Solutions:
Graph
the related function f(x) = x2 – 10x +
16. Before making a table of values, find the equation of the axis of symmetry.
This will make selecting x-values for your table easier.
The
equation of the axis of symmetry is x = 5. Now, make a table using x-values
around 5. Graph each point on a coordinate plane.
The
zeros of the function appear to be 2 and 8. So, the roots are 2 and 8.
Sometimes exact roots cannot be found
by graphing. In this case, estimate solutions by stating the consecutive
integers between which the roots are located.
Example
3.
Estimate the roots of –x2 + 2x + 1 = 0.
Alternative Solutions:
Find
the equation of the axis of symmetry.
The
equation of the axis of symmetry is x = 1. Now, make a table using x-values
around 1. Graph each point on a coordinate plane.
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The x-intercepts
of the graph are between 1 and 0 and between 2 and 3. So, one root of the
equation is between 1 and 0, and the other root is between 2 and 3.
Example
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4.
Find
two numbers whose sum is 4 and whose product is 5.
Alternative Solutions:
The graphing calculator is a useful tool when making tables
in order to solve functions.
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