Transcript Completing the Square

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To review how to solve quadratic equations in
vertex form
To solve quadratic equations in general form
by completing the square
• Recall that rectangle diagrams help you
factor some quadratic expressions.
• How do you find the roots of an equation such as
0=x2+x-1? It is not a perfect-square trinomial,
nor is it easily factorable. For these equations
you can use a method called completing the
square.
Searching for
Solutions
• To understand
how to complete
the square with
quadratic
equations, you’ll
first work with
rectangle
diagrams.
• Complete each
rectangle diagram
so that it is a
square. How do
you know which
number to place
in the lower-right
corner?
Searching for
Solutions
• For each diagram,
write an equation
in the form
x2+bx+c=(x+h)2.
• On which side of
the equation can
you isolate x by
undoing the order
of operations?
9
4
9
6.25
b2
4
Searching for
Solutions
• Suppose the
area of each
diagram is 100
square units.
• For each square,
write an
equation that
you can solve
for x by undoing
the order of
operations.
• Solve each
equation
symbolically.
You will get two
values for x.
4
9
6.25
• The solutions for x in the equations
from the last slide are rational
numbers. This means you could have
factored the equations with rational
numbers. However, the method of
completing the square works for other
numbers as well. Next you’ll consider
the solution of an equation that you
cannot factor with rational numbers.
• Consider the
equation
x2+6x-1=0.
Describe
what’s
happening in
each stage of
the solution
process.
• Use your calculator to find decimal
approximations for 3  10 and 3  10 .
• Then enter the equation y=x2+6x-1 into
Y1. Use a calculator graph or table to
check that your answers are the xintercepts of the equation.
• Repeat the solution stages on to find the
solutions to x2+8x-5=0.
• The key to solving by
completing the square is to
express one side of the
equation as a perfectsquare trinomial.
• In the investigation the
equations are in the form
y=1x2+bx+c. Note that the
coefficient of x2, called the
leading coefficient, is 1.
However, there are other
perfect-square trinomials.
• An example of perfectsquare trinomial with a
different leading coefficient
is shown at right.
Example A
• Solve the equation 3x2+18x-8=22 by completing
the square.
• First, transform the equation so that you can
write the left side as a perfect-square trinomial
in the form x2+2hx+h2.
• Now you need to decide what number to
add to both sides to get a perfect-square
trinomial on the left side.
• Use a rectangle diagram to make a
square. When you decide what number to
add, you must add it to both sides to
balance the equation.
The two solutions are 3  19 , or approximately
1.36, and 3  19 , or approximately -7.36.
Example B
• Find the vertex form of the equation
y =2x2+12x+21.
• Then identify the vertex and any x-intercepts of
the parabola.
• To convert y =2x2+12x+21 to the form
y=a(x-h)2+k, complete the square.
• Now you can complete the square on the
expression inside the parentheses.