Solve Quadratic Equations with the Quadratic Formula

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Solve Quadratic Equations with the Quadratic
Formula

• Section 9.2
• MATH 116-460
• Mr. Keltner

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The Quadratic Formula
If we have a quadratic equation in the form ax2
bx c 0 and a ? 0, we can derive a formula
that can be used to solve any quadratic equation
in standard form. By completing the square on the
quadratic equation ax2 bx c 0, we get
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Example 1

• Solve x2 - 5x 7.

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Example 2

• Solve 16x2 - 23x 17x - 25.

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Example 3

• Solve x2 - 6x 10 0.

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Using the Calculator to Simplify
To find solutions using the calculator, it is
very important to group quantities together using
parentheses. You may push 2nd-ENTER to recall the
last entry you made, then change to a subtraction
sign before the square root symbol. This finds
both approximate solutions using a calculator.
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The Discriminant

• In the Quadratic Formula, the expression b2-4ac
  is called the disciminant of the equation of the
  form ax2 bx c 0.

Value of b2-4ac Number of Solutions
b2-4ac gt 0 (Positive) TWO real solutions
b2-4ac 0 (Zero) ONE real solutions
b2-4ac lt 0 (Negative) ZERO real solutions (TWO imaginary solutions)
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The Discriminant vs.the Graph of the Function
b2-4ac gt 0 (Positive)
b2-4ac 0 (Zero)
b2-4ac lt 0 (Negative)

• If we graph the function y ax2 bx c, the
  number of x-intercepts corresponds to the number
  or real solutions to the equation ax2 bx c
  0.

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Example 4

• Find the discriminant of the quadratic equation
  and give the number and type of solutions of the
  equation.
• x2 10x 23 0
• x2 10x 25 0
• x2 10x 27 0

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Example 5

• Find the x- and y-intercepts of each function.
• You may verify your answers on a graphing
  calculator.

• y 3x2 - 5x 1
• y x2 - 8x 16

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Example 6

• A basketball player passes the ball to a
  teammate. The ball leaves the players hand 5
  feet above the ground and has an initial vertical
  velocity of 55 feet per second. The teammate
  catches the ball when it returns to a height of 5
  feet. How long was the ball in the air?

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Assessment

• Pgs. 642-644
• s 7-84, multiples of 7