Quadratic Equations

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Chapter 7

• Quadratic Equations

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• A quadratic equation is one that can be written
  in the form _____________________________ where
  a, b, c are real numbers and a ?0.
• The degree of a quadratic equation is ______.
• E.g.

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• The related function has equation ______________
• It has a graph in the shape of a _______________.
• Every quadratic equation has two solutions
  (roots). They may be
• a) ___________________________________
• b) ___________________________________
• c) ___________________________________

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Methods of Solving Quadratic Equations

• Graphing Graph related function
  and locate its real roots
  (x-intercepts)
• On TI-83/84, use 2nd Calc 2 Zero
• Factoring If possible, factor the expression.
  Set each factor equal to zero and solve.
• Quadratic Formula

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Solve by graphing on the calculator. Give answers
to nearest tenth.
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Solve by factoring.
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Solve by factoring.
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Solve by factoring.
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Solve the higher order equation by factoring.
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Solve the higher order equation by factoring.
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Solve using the Quadratic Formula.
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Solve using the Quadratic Formula (Give answers
to two decimal places.)
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• For equations in the form
  , the discriminant is the value of
  ______________ .(This is the expression under
  the radical in the Quadratic Formula.)

We can use the discriminant to determine the
character (number and type) of the roots of a
quadratic equation.
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Character of the Roots

• If b2 4ac gt 0 and is a perfect square, the
  equation has _____________________________________
  ___.
• If b2 4ac gt 0 and is NOT a perfect square, the
  equation has _____________________________________
  __.
• If b2 4ac 0, the equation has
  __________________________________________.
• If b2 4ac lt 0, the equation has
  ________________
• ___________________________.

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First find the value of the discriminant then
use it to describe the number and type of roots.
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Mathematical Modeling

• In real world applications we often encounter
  numerical data in the form of a table. The
  powerful mathematical tool, regression analysis,
  can be used to analyze numerical data. In
  general, regression analysis is a process for
  finding a function that best fits a set of data
  points.
• In the next example, we use a linear model
  obtained by using linear regression on a graphing
  calculator.

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Regression Notes

• Regression a process used to relate two
  quantitative variables.
• Independent variable the x variable (or
  explanatory variable)
• Dependent variable the y variable (or response
  variable)
• To interpret the scatterplot, identify the
  following
• Form
• Direction (for linear models)
• Strength

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Form

• Form the function that best describes the
  relationship between the two variables.
• Some possible forms would be linear, quadratic,
  cubic, exponential, or logarithmic.

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Direction

• Direction a positive or negative direction can
  be found when looking at linear regression lines
  only.
• The direction is found by looking at the sign of
  the slope.

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Strength
Strength how closely the points in the data are
gathered around the form.
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Making Predictions

• Predictions should only be made for values of x
  within the span of the x-values in the data set.
• Predictions made outside the data set are called
  extrapolations, which can be dangerous and
  ridiculous thus, extrapolating is not
  recommended.

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Example of Linear Regression
Prices for emerald-shaped diamonds taken from an
on-line trader are given in the following table.
Find the linear model that best fits this
data. Weight (carats) Price 0.5 1,677 0.6 2,
353 0.7 2,718 0.8 3,218 0.9 3,982
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Scatter Plots
Enter these values into the lists in a graphing
calculator as shown below .
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Scatter Plots
We can plot the data points in the previous
example on a Cartesian coordinate plane, either
by hand or using a graphing calculator. If we
use the calculator, we obtain the following plot
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Example of Linear Regression(continued)
Based on the scatterplot, the data appears to be
linearly correlated thus, we can choose linear
regression from the statistics menu, we obtain
the second screen, which gives the equation of
best fit.
The linear equation of best fit is y 5475x -
1042.9.
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Scatter Plots
We can plot the graph of our line of best fit on
top of the scatterplot
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Making a Prediction

• Is it appropriate to use the model to predict the
  price of an emerald-shaped diamond that weighs
  0.75 carats? If so, estimate the price.
• Is it appropriate to use the model to predict the
  price of an emerald-shaped diamond that weighs
  2.7 carats? If so, estimate the price.

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Quadratic Regression
A visual inspection of the plot of a data set
might indicate that a parabola would be a better
model of the data than a straight line. In
that case, rather than using linear regression to
fit a linear model to the data, we would use
quadratic regression on a graphing calculator to
find the function of the form y ax2 bx c
that best fits the data. From the ?? CALC menu,
choose 5 QuadReg
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Example of Quadratic Regression
An automobile tire manufacturer collected the
data in the table relating tire pressure x (in
pounds per square inch) and mileage (in thousands
of miles.) x Mileage 28 45 30 52 32 55 34 51 36 47
Using quadratic regression on a graphing
calculator, find the quadratic function that best
fits the data. Round values to 6 decimal places.
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Example of Quadratic Regression(continued)
Enter the data in a graphing calculator and
obtain the lists below.
Choose quadratic regression from the STAT Calc
menu and obtain the coefficients as shown
This means that the equation that best fits the
data is y -0.517857x2 33.292857x-
480.942857
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Example of Quadratic Regression(continued)
Use the model to estimate the number of miles you
could get from tires inflated at a) 35 psi and
b) 40 psi.
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Another Example of Modeling
The following table shows crop yields, Y (in
bushels), for various amounts of fertilizer used,
x (in lbs/100 ft2), for 18 different equally
sized plots.

Plot 1 2 3 4 5 6 7 8 9 19 11 12 13 14 15 16 17 18
x Fertilizer (lbs/ 100ft2) 0 0 5 5 10 10 15 15 20 20 25 25 30 30 35 35 40 40
Y Yield (bushels) 4 6 10 7 12 10 15 17 18 21 20 21 21 22 21 20 19 19
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Example (continued)

1. Use your calculator to graph a scatter plot of
   the data and comment on the type of relationship
   that exists between the two variables (the amount
   of fertilizer used , x, and the crop yield, y.)

It appears that the data follows a quadratic
relationship with a lt 0.
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Example (continued)

1. Use the calculator to find the quadratic function
   of best fit. Give values to 4 significant digits.
   Sketch this function in the same window as your
   scatter plot.

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Example (continued)

1. Use the function to predict the optimal amount of
   fertilizer (in pounds per 100ft2) to use and the
   crop yield (in bushels) when the optimal amount
   of fertilizer is applied. Give values to 3
   significant digits.

Use the graphing calculator and the graph of the
quadratic model to find the maximum point.
According to the model, if we apply 31.5 pounds
of fertilizer per 100 sq. feet, the crop yield
will be 20.8 bushels.