1.5 Quadratic Equations

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1.5 Quadratic Equations
Start p 145 graph and model for 131 discuss.
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Definition of a Quadratic Equation

• A quadratic equation in x is an equation that can
  be written in the standard form
• ax2 bx c 0
• where a, b, and c are real numbers with a not
  equal to 0. A quadratic equation in x is also
  called a second-degree polynomial equation in x.

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The Zero-Product Principle

• If the product of two algebraic expressions is
  zero, then at least one of the factors is equal
  to zero.
• If AB 0, then A 0 or B 0.
• Q Will this work for any other number, such as
  AB5?

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Solving a Quadratic Equation by Factoring

1. If necessary, rewrite the equation in the form
   ax2 bx c 0, moving all terms to one side,
   thereby obtaining ______ on the other side.
2. Factor.
3. Set each factor zero. (Apply the zero-product
   principle.)
4. Solve the equations in step 3.
5. Check the solutions in the __________ equation.

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Text Example

• Solve 2x2 7x 4 by factoring and then using
  the zero-product principle. (Do not look at
  notes, no need to write.)
• Step 1 Move all terms to one side and obtain
  zero on the other side. Subtract 4 from both
  sides and write the equation in standard form.
• 2x2 7x - 4 4 - 4
• 2x2 7x - 4 0
• Step 2 Factor.

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Solution cont.

• Solve 2x2 7x 4 by factoring and then using
  the zero-product principle.
• Steps 3 and 4 Set each factor equal to zero
  and solve each resulting equation.
• 2 x - 1 0 or x 4 0
• 2 x 1 x -4
• x 1/2
• Steps 5 check your solution (by putting each
  solution back into the ORIGINAL equation to see
  if it yields a TRUE statement.

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• Ex Solve for x
• (2x -3)(2x 1) 5 Why cant we set each
  factor 5?
• ALWAYS begin by
• factoring out the GCF.

Simplify Set 0 Factor Apply zero product
principle Check.
Q In the above example, it is not necessary to
set the factor 4 0, but what if the GCF had
been 4x?
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The Square Root Method

• If u is an algebraic expression and d is a
  positive real number, then u2 d has exactly two
  solutions.
• If u2 d, then u or u -
• Equivalently,
• If u2 d then u ?
• We only use this method if the variable is
  originally contained within a squared part.
  Ex x2-812 or (2x-4)2 520. Can you think of
  a counter example? Do

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Text Example

• What term should be added to the binomial
• x2 8x so that it becomes a perfect square
  trinomial? Then write and factor the trinomial.
• x2 8x ____2
• (x ____)2
• Note this is still an expression, not an
  equation.
• Do (factor by completing the square- see
  instructions next slide first) p 144 54.

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Completing the Square

• If x2 bx is a binomial, then by adding (b/2) 2,
  which is the square of half the coefficient of x,
  a perfect square trinomial will result. That is,
• x2 bx (b/2)2 (x b/2)2
• That is, take half of the coefficient of the x
  term, square it, and add it to each side.
• Then take /- the square root of each side.
  (Square root method.)
• Note this is really just using the fact that
  the square of a binomial results in a perfect
  square trinomial. We are just completing the
  perfect square trinomial.

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Solving by the Quadratic Formula
Given a quadratic equation in the form
agt0, a,b,c integers
We can solve for x by plugging in a, b and c
Derived by completing the square, if interested,
see p121.
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Ex Solve by using the quadratic formula
Put into form Identify a,b,c Plug in Simplify
Common errors Not writing the division bar all
the way. -b means (whatever b is!). In this
case (-8) 8.
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The Discriminant and the Kinds of Solutions to
ax2 bx c 0
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Which approach do we use to solve a quadratic
equation?

• 1. Recognize that you have a quadratic equation.
• If the variable is isolated within the squared
  part, isolate the squared part, take /- square
  root of each side, then isolate the variable.
  (Square root method.)
• Otherwise set 0
• a. If it is EASY to factor, factor, set each
  factor equal to zero and solve for the variable
  (Factoring method.)
• b. If it is NOT easy to factor, plug a, b, and
  c into the quadratic formula and simplify
  (Quadratic formula method.)
• 4. If it says to solve by completing the square,
  do so (Completing the square method.)

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The Pythagorean Theorem

• The sum of the squares of the lengths of the legs
  of a right triangle equals the square of the
  length of the hypotenuse.
• If the legs have lengths a and b, and the
  hypotenuse has length c, then
• a2 b2 c2 do p144105, 138 (set up)