Quadratic Equations, Functions, and Models

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Section 2.3

• Quadratic Equations, Functions, and Models

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Quadratic Equations- second degree equations of
a single variable (highest
power of variable is 2)Quadratic Equations can
have at most 2 real solutions.
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Quadratic Equation

• Quadratic Equation
• Standard Form
• ax2 bx c 0
• where a, b, c are real numbers and a ? 0.
• Quadratic Function
• f(x) ax2 bx c
• where a, b, c are real numbers and a ? 0.

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Strategies for Solving a Quadratic Equations

• Factoring (Zero-Product Property)
• Square Root Property
• Completing the Square
• 4. Quadratic Formula

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Zero-Product Property

• If the product of two numbers is zero (0), then
  one of the numbers is zero (0).
• ab 0 ,
• where a and b are real numbers
• a or b must be zero

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Steps for Solving Quadratic Equations by
Factoring (Zero-Product Property)

• Set quadratic equal to zero.
• ax2 bx c 0 , where a, b, c are real
  numbers and a ? 0
• Factor.
• Set each factor equal to zero.
• Solve each equation for the variable.

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Example of Solving a Quadratic Equation

• ex. Solve for x.
• x2 2x 15 0

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Graph of f(x) x² 2x - 15
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Zeros of a Function

• The zeros of a quadratic function
• f(x) ax2 bx c 0 are the solutions of the
  associated quadratic equation
• ax2 bx c 0. (These solutions are
  sometimes called the roots of the equation.)
• Real number zeros (solutions) are the
  x-coordinates of the x-intercepts of the graph of
  the quadratic equation.

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Zeros of a Function

• When f(x) 0, then you are finding the
• the zero(s) of the function.
• f(x) 0 means y 0
• Which means we are finding the x-intercept(s)
• Zero of a function is another name for
  x-intercept
• Zero roots solutions x-intercepts

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Solving Quadratic Equations with the Square Root
Property

• x2 k
• Examples

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Completing the Square

• 1.Isolate the terms with variables on one side of
  the equation and arrange them in descending
  order.
• 2. Divide by the coefficient of the squared term
  if that coefficient is not 1.
• 3. Complete the square by taking half the
  coefficient of the first-degree term and adding
  its square on both sides of the equation.
• 4. Express one side of the equation as the square
  of a binomial.
• 5. Use the principle of square roots.
• 6. Solve for the variable.

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Steps for Solving Quadratic Equations by Using
Quadratic Formula

• Quadratic Equation
• ax2 bx c 0 , where a, b, c are real
  numbers and a ? 0
• Quadratic Formula

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Discriminant

• b²- 4ac
• If the value of the discriminant is positive,
  then there are 2 real solutions.
• If the value of the discriminant is zero, then
  there is 1 real solution.
• If the value of the discriminant is negative,
  then there are 2 imaginary solutions.