5) Solving Quadratic Equations (1)

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1-5 Solving Quadratic Equations using Completing
the square and Quadratic formula

• A-REI.4a - complete the square to transform any
  quadratic equation in x into (x-p)2 q that has
  same solutions.
• A-REI.4b - Solve, find the roots, the zeros, for
  quadratic equations in 1 variable by sq rt,
  factoring, quadratic formula
• and completing the square.
  Recognize when quadratic formula give complex
  solutions, write correctly.
• N-CN.7 --Solve quadratic equations with real
  coefficients, and complex solutions
• F-IF.8a.- Use the process of factoring and
  completing the square in a quadratic function to
  show zeros, extreme values,
• and symmetry of the graph, and
  interpret these in terms of a context.

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Completing the Square is based on Solving by
Square Roots

• Get the squared variable or quantity alone
• Square Root both sides

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Perfect Square Trinomials

• ax2 bx c 0
• a and b term are perfect squares
• 2ab b
• a2 2ab b2 (a b)2
• a2 2ab b2 (a b)2

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Solve these perfect square trinomials
This side is a perfect square trinomial
Rewrite it in its factored form
Now, square root both sides
Simplify
Solve for x
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Rewrite the expression as a perfect square
trinomial

• What would c have to be, to make these perfect
  square trinomials?

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Complete the square, and show what the perfect
square trinomial is, then factor it
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If a quadratic equation is not a perfect square
trinomial, turning it into one is called
completing square

• 1. rewrite the equation into 1x2 bx c
• 2. complete the square, add (b/2)2 to both sides
  of
• 3. factor the perfect square trinomial
  (x )2 c (b/2)2
• 4. square root both sides and solve for x

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Steps to solve by completing the square

• 1.) If the quadratic does not factor, move the
  constant to the other side of the equation
• Ex x²-4x -7 0 x²-4x7
• 2.) Work with the x² x side of the equation and
  complete the square by taking ½ of the
  coefficient of x and squaring
• Ex. x² -4x 4/2 2²4
• 3.) Add the number you got to complete the square
  to both sides of the equation
• Ex x² -4x 4 7 4
• 4.)Simplify your trinomial square
• Ex (x-2)² 11
• 5.)Take the square root of both sides of the
  equation
• Ex x-2 v11
• 6.) solve for x
• Ex x2v11

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Solve by Completing the Square
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Solve by Completing the Square
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Solve by Completing the Square
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Solve by Completing the Square
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Solve by Completing the Square
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Solve by Completing the Square
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The coefficient of x2 must be 1
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THE QUADRATIC FORMULA

• When you solve using completing the square on
  the general formula you get
• identify a, b, and c then substitute into the
  formula and solve for x.
• Best used when equation is not factorable

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WHY USE THE QUADRATIC FORMULA?

• The quadratic formula allows you to solve ANY
  quadratic equation, even if you cannot factor it.
• An important piece of the quadratic formula is
  whats under the radical
• b2 4ac
• This piece is called the discriminant.

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WHY IS THE DISCRIMINANT IMPORTANT?

• The discriminant tells you the number and types
  of answers
• (roots) you will get. The discriminant can be ,
  , or 0
• which actually tells you a lot! Since the
  discriminant is
• under a radical, think about what it means if you
  have a
• positive or negative number or 0 under the
  radical.

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WHAT THE DISCRIMINANT TELLS YOU!
Value of the Discriminant Nature of the Solutions
Negative 2 imaginary solutions
Zero 1 Real Solution
Positive perfect square 2 Reals- Rational
Positive non-perfect square 2 Reals- Irrational
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Find the value of the discriminant and describe
the nature of the roots (real,imaginary,
rational, irrational) of each quadratic equation.
Then solve the equation using the quadratic
formula) 1.
a2, b7, c-11
Discriminant
Value of discriminant137 Positive-NON perfect
square Nature of the Roots 2 Reals - Irrational
Discriminant
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Example 1- continued
Solve using the Quadratic Formula
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Quadratic Formula (Example 2)

• x2 8x 17 0
• a 1
• b 8
• c 17

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• 3. 3x28x35
• 3x28x-350
• a3, b8, c -35

OR
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• 4. -2x2-2x3
• -2x22x-30
• a-2, b2, c -3